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underline="false"/><Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" name="_cstyle29" readonly="false" size="12" underline="false"/><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" name="_cstyle28" readonly="false" size="12" underline="false"/><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" name="_cstyle27" readonly="false" size="12" underline="false"/><Font background="[0,0,0]" bold="true" executable="false" family="Monospaced" foreground="[255,0,0]" italic="false" name="_cstyle25" readonly="false" size="12" underline="false"/><Font background="[0,0,0]" bold="true" executable="false" family="Monospaced" foreground="[255,0,0]" italic="false" name="_cstyle24" readonly="false" size="14" underline="false"/><Font background="[0,0,0]" bold="true" executable="false" family="Times New 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readonly="false"/><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" name="_pstyle20" readonly="false" size="12" underline="false"/></Styles><Page-Numbers enabled="false" first-number="1" first-numbered-page="1" horizontal-location="right" style="Page Number" vertical-location="bottom"/><Text-field layout="_pstyle11" style="_cstyle18">Homegeneous first order differential equations</Text-field><Text-field layout="_pstyle12" style="_cstyle19">by Peter Stone, Dept. of Applied and Environmental Sciences, RMIT</Text-field><Text-field layout="_pstyle12" style="_cstyle19">peter.stone@rmit.edu.au . . or . . peterstone@optusnet.com.au</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">Version:  10.12.2003</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">restart;</Text-field></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section><Title><Text-field layout="_pstyle14" style="ParagraphStyle4"><Font style="_cstyle22">load </Font><Font style="_cstyle24">desolve</Font></Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">RMIT file path to read Maple m-file of differential equation solving procedures with the interface </Font><Font style="_cstyle25">desolve</Font><Font style="_cstyle19">. </Font></Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">read "J:\\Class_Notes/Peter Stone/MapleMath/procdrs/DEsol.m";</Text-field></Input><Output><Text-field layout="_pstyle13" style="_cstyle21">Error, could not open `J:\Class_Notes/Peter Stone/MapleMath/procdrs/DEsol.m` for reading
</Text-field></Output></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">Another file path. </Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">read "D:\\Maple 9/procedures/DEsol.m";</Text-field></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Solving 1st order homogeneous differential equations</Text-field></Title><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">The procedure </Font><Font style="_cstyle25">dsolveHG</Font><Font style="_cstyle19"> in the next section tries to solve a differential equation of the form</Font></Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27"> </Font><Equation input-equation="dy/dx = F(y/x);" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiRkc2IyomJSJ5R0YmJSJ4R0Yo</Equation><Font style="_cstyle27">    </Font></Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">by means of the substitution </Font><Equation input-equation="y = v*x;" style="2D Comment">NiMvJSJ5RyomJSJ2RyIiIiUieEdGJw==</Equation><Font style="_cstyle19">  .</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">Since </Font><Equation input-equation="dy/dx = v+x;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYlInZHRiYlInhHRiY=</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dv/dx;" style="2D Comment">NiMqJiUjZHZHIiIiJSNkeEchIiI=</Equation><Font style="_cstyle19">  , the differential equation becomes </Font><Equation input-equation="v+x;" style="2D Comment">NiMsJiUidkciIiIlInhHRiU=</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dv/dx = F(v);" style="2D Comment">NiMvKiYlI2R2RyIiIiUjZHhHISIiLSUiRkc2IyUidkc=</Equation><Font style="_cstyle19">  .</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">This equation can be written in the form</Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27"> </Font><Equation input-equation="x;" style="2D Comment">NiMlInhH</Equation><Font style="_cstyle27">   </Font><Equation input-equation="dv/dx = F(v)-v;" style="2D Comment">NiMvKiYlI2R2RyIiIiUjZHhHISIiLCYtJSJGRzYjJSJ2R0YmRi1GKA==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">or</Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  </Font><Equation input-equation="1/(F(v)-v);" style="2D Comment">NiMqJiIiIkYkLCYtJSJGRzYjJSJ2R0YkRikhIiJGKg==</Equation><Font style="_cstyle27">   </Font><Equation input-equation="dv/dx = 1/x;" style="2D Comment">NiMvKiYlI2R2RyIiIiUjZHhHISIiKiZGJkYmJSJ4R0Yo</Equation><Font style="_cstyle27">    </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">and so has separable variables.</Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">Integrating both sides with respect to </Font><Font style="_cstyle28">x</Font><Font style="_cstyle19"> gives</Font></Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  </Font><Equation input-equation="Int(1/(F(v)-v),v) = Int(1/x,x);" style="2D Comment">NiMvLSUkSW50RzYkKiYiIiJGKCwmLSUiRkc2IyUidkdGKEYtISIiRi5GLS1GJTYkKiZGKEYoJSJ4R0YuRjI=</Equation><Font style="_cstyle27">  , </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">so  </Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">    </Font><Equation input-equation="ln(x) = Int(1/(F(v)-v),v);" style="2D Comment">NiMvLSUjbG5HNiMlInhHLSUkSW50RzYkKiYiIiJGLCwmLSUiRkc2IyUidkdGLEYxISIiRjJGMQ==</Equation><Font style="_cstyle27">  .</Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">If this integral can be found analytically, we shall obtain at least an implicit description of the solution.</Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">To suggest that we need to replace </Font><Font style="_cstyle28">v</Font><Font style="_cstyle19"> in the result by </Font><Equation input-equation="y(x)/x;" style="2D Comment">NiMqJi0lInlHNiMlInhHIiIiRichIiI=</Equation><Font style="_cstyle19">  , we could rewrite the previous equation in the form</Font></Text-field><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">   </Font><Equation input-equation="ln(x) = Int(1/(F(v)-v),v = `` .. y(x)/x);" style="2D Comment">NiMvLSUjbG5HNiMlInhHLSUkSW50RzYkKiYiIiJGLCwmLSUiRkc2IyUidkdGLEYxISIiRjIvRjE7JSFHKiYtJSJ5R0YmRixGJ0Yy</Equation><Font style="_cstyle27">  . </Font></Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">In particular cases it may be possible to solve for </Font><Equation input-equation="y(x)" style="2D Comment">NiMtJSJ5RzYjJSJ4Rw==</Equation><Font style="_cstyle19">   explicitly.</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/></Section><Section><Title><Text-field layout="_pstyle14" style="ParagraphStyle4"><Font style="_cstyle22">A procedure for solving homogeneous 1st order DE's: </Font><Font style="_cstyle24">desolveHG</Font></Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">desolveHG: usage</Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle29">Calling Sequence:</Font>
</Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle29">  </Font><Font style="_cstyle19">   desolveHG( de )
     desolveHG( {de,ic} )
     desolveHG( {de,ic},y(x) )</Font>
</Text-field><Text-field layout="_pstyle17" style="_cstyle30">Parameters:</Text-field><Text-field layout="_pstyle12" style="_cstyle19">    </Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle31">   de   - </Font><Font style="_cstyle19">     a first order differential equation with the derivative given in the form diff(y(x),x),</Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">                        (if x and y are the independent and dependent variables respectively).</Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle31">   ic   - </Font><Font style="_cstyle19">     an initial condition in the form y(x0) = y0. </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">         </Text-field><Text-field layout="_pstyle17" style="_cstyle30">Description:</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">The procedure </Font><Font style="_cstyle25">desolveHG</Font><Font style="_cstyle19"> attempts to solve a first order differential equation by making a substitution of the form:  </Font><Equation input-equation="y = _v*`.`*x;" style="2D Comment">NiMvJSJ5RyooJSNfdkciIiIlIi5HRiclInhHRic=</Equation><Font style="_cstyle19">  .</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">If no initial condition is given, a general solution with an arbitrary constant is sought.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle29">Options:</Font>
</Text-field><Text-field layout="_pstyle12" style="_cstyle19">info=true/false
The option info=true causes the intermediate steps in the solution to be printed.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">format=explicit/implicit</Text-field><Text-field layout="_pstyle12" style="_cstyle19">When format=explicit, which is the default option, an attempt is made to solve for y explicitly.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/></Section><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle32">How to activate:</Font><Font style="_cstyle19">
To make the procedure active open the subsection, place the cursor anywhere after the prompt [ &gt;  and press [Enter].
You can then close up the subsection.</Font></Text-field><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">desolveHG: implementation </Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">desolveHG := proc(ff)
   
   local vars,derivs,x,yx,y,df,dff,e1,intg,fv,_v,fvar,
         prntflg,Options,drv,val,rs,islog,i,j,lsic,sol,
         soln,soln2,sols,initcond,de,ic,x0,y0,rsic,const,
         exprs,assignedconst,goodsols,frmt,pm,rightOK,lc,
         const2,separate,exp2,islg,startopts,xx,yy,ee,cc,yy0,
         Cs,gt,jS,j1,j2,s,nvars;
   global C;
   
   # Remove any solutions involving "RootOf".
   goodsols := proc(sol::list)
      local goodlist,i;
      goodlist := NULL;
      for i from 1 to nops(sol) do
         if indets(sol[i],'specfunc(anything,RootOf)')={} then
            goodlist := goodlist,sol[i];
         end if;
      end do;
      [goodlist];           
   end proc:
   
    # Separate into two factors independent and dependent of x  
    separate := proc(f::algebraic,x::symbol)
 
      local cansep,opr,np,i,xops,cops,xps,cps,fx,fc;

      if op(0,f)&lt;&gt;`*` then
         if member(x,indets(f,name)) then
            return [1,f]
         else
            return [f,1]
         end if;
      end if;

      opr := [op(f)];
      np := nops(opr);

      # Split into 2 lists of factors
      xops := NULL: cops := NULL:
      for i from 1 to np do
         if member(x,indets(opr[i],name)) then
            xops := xops,opr[i];
         else
            cops := cops,opr[i];
         end if;
      end do;
      xps := [xops]:
      cps := [cops]:
      fx := mul(xps[i],i=1..nops(xps));
      fc := mul(cps[i],i=1..nops(cps));
      [fc,fx];
   end proc: # of separate

   # Apply exp giving precedence to expressions in x.
   exp2 := proc(fx::algebraic,x::symbol)
      local sep,i,ff;
      ff := expand(fx);
      if op(0,ff)=`+` then
         return mul(exp2(op(i,ff),x),i=1..nops(ff))
      end if;
      if op(0,ff)=`*` then
         sep := separate(ff,x);
        if type(sep[2],function) and op(0,sep[2])='ln' 
              and nops(sep[2])=1 and type(op(1,sep[2]),algebraic) then
            return op(1,sep[2])^sep[1];
         end if;
      end if;
      if type(ff,function) and op(0,ff)='ln' 
              and nops(ff)=1 and type(op(1,ff),algebraic) then
        return op(1,ff);
      end if;
      return simplify(exp(ff));  
   end proc: # of exp2

   # Check whether fx has form of a sum of logs of expressions in x
   islog := proc(fx::algebraic,x::symbol)
      local ff,i,opf,sep,islg;
      islg := true; 
      ff := expand(fx); 
      if op(0,ff)=`+` then
         opf := [op(ff)];
         islg := true;
         for i from 1 to nops(ff) do
            sep := separate(opf[i],x);
            if not type(sep[2],function) or not op(0,sep[2])='ln' 
              or nops(sep[2])&lt;&gt;1 or not type(op(1,sep[2]),algebraic) then
               islg := false;
               break;
            end if;
         end do;
      elif type(ff,function) and op(0,ff)=`ln` and nops(ff)=1 then
         islg := evalb(type(op(1,ff),algebraic)
              and member(x,indets(op(1,ff),name)));
      elif op(0,ff)=`*` then
         sep := separate(ff,x);
         islg := evalb(type(sep[2],function) and op(0,sep[2])='ln' 
              and nops(sep[2])=1 and type(op(1,sep[2]),algebraic));
      else islg := false
      end if;
      islg;
   end proc: # of islog
   
   # start of main procedure
   initcond := false;
   if type(ff,set(equation)) and nops(ff)=2 then
      de := op(1,ff);
      ic := op(2,ff);
      if not has(de,diff) then
         de := op(2,ff);
         ic := op(1,ff);
      end if;
      initcond := true;
   elif type(ff,equation) then de := ff
   else
      error "the 1st argument, %1, is invalid .. it should be an equation or a set of 2 equations"
   end if;

   startopts := 2;
   if nargs&gt;1 then
      ee := args[2];
      if type(ee,function) and nops(ee)=1 then
         yy := op(0,ee);
         xx := op(1,ee);
         if type(xx,name) and type(yy,name) then
            startopts := 3;
         else
            error "the 2nd argument, %1, has incorrect form for the dependent variable",ee;
         end if;
      end if;
   end if;

   prntflg := false;
   frmt := 'explicit';
   if nargs&gt;1 then
      Options:=[args[startopts..nargs]];
      if not type(Options,list(equation)) then
         error "each optional argument must be an equation"
      end if;
      if hasoption(Options,'info','prntflg','Options') then 
         if prntflg&lt;&gt;true then prntflg := false end if;
      end if;
      if hasoption(Options,'format','frmt','Options') then
         if not (frmt='explicit' or frmt='implicit') then
            error "\"format\" must be 'explicit' or 'implicit'"
         end if;
      end if;
      if nops(Options)&gt;0 then
         error "%1 is not a valid option for %2",op(1,Options),procname;
      end if;
   end if;

   if has(de,'D') then de := convert(de,'diff') end if;
   derivs := indets(de,'specfunc(anything,diff)');
   if derivs={} then
      error "the equation, %1, is not an ordinary differential equation",de;
   end if;
   nvars := nops(indets(derivs,'name'));
   if nvars&lt;&gt;1 then
      if nvars=0 then
         error "there is a problem with the independent variable occurring in the derivative(s)";
      else
         error "there should only be one independent variable in the differential equation"
      end if;
   end if;
   nvars := nops(indets(derivs,'anyfunc(name)'));
   if nvars&lt;&gt;1 then
      if nvars=0 then
         error "there is a problem with the dependent variable occurring in the derivative(s)"
      else
         error "there should only be one dependent variable in the differential equation"
      end if;
   end if;

   if nops(derivs)&lt;&gt;1 then
      error "there are too many derivatives in the differential equation .. note that the differential equation must be of order 1"
   end if;
   df := op(1,derivs);
   if type(df,function) and op(0,df)=diff and nops(df)=2 then
      yx := op(1,df);
      if not type(yx,anyfunc(name)) then
         error "the 1st argument %1, in the derivative, %2, is invalid .. it should be the 'unknown' dependent variable",yx,df;
      end if; 
      x := op(2,df);
      if not type(x,name) then
         error "the 2nd argument %1, in the derivative, %2, is invalid .. it should be the dependent variable",x,df;
      end if; 
   else
      error "the derivative, %1, does not make sense",df;
   end if;

   y := op(0,yx);
   vars := indets(de,name);
   if member(y,vars) then
      error "%1 and %2 cannot both appear in the differential equation",yx,y;
   end if;
   if op(1,yx)&lt;&gt;x then
      error "the derivative, %1, does not make sense",df;
   end if;

   if startopts=3 then 
      if x&lt;&gt;xx or y&lt;&gt;yy then
         error "cannot solve the differential equation for %1",ee;
      end if;
   end if;
 
   if initcond then
      lsic := lhs(ic);
      if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 
                             and type(op(1,lsic),algebraic) then
         x0 := op(1,lsic);
         if has(x0,{x,y}) then
            error "initial condition must not involve %1 or %2",x,y;
         end if;
      else
         error "initial condition is not decipherable"
      end if;
      rsic := rhs(ic);
      if type(rsic,algebraic) then
         y0 := rsic;
         if has(y0,{x,y}) then
            error "initial condition must not involve %1 or %2",x,y;
         end if;
      else
         error "initial condition is not decipherable"
      end if;
   end if;

   drv := solve(de,df);
   if nops([drv])=1 then
      e1 := subs(yx=y,factor(drv));
   else
      error "cannot obtain a unique expression for the derivative"
   end if;

   if assigned(C) and not type(eval(C),table) then
      C := table();
      WARNING("C has been redefined as a table for use as arbitrary constants");
   end if;
   # Find the indices j1,j2 to use in the constants
   Cs := select(type,indets(de),'specindex(posint,C)');
   gt := proc(_u) local s,j;
            typematch(_u,C[j::posint],'s'); 
            subs(s,j)
         end proc:
   jS := sort([op(map(gt,Cs))]);
   for i to nops(jS)+1 do
      if not member(i,jS) then j1 := i; break; end if;   
   end do;
   jS := sort([j1,op(jS)]);
   for i to nops(jS)+1 do
      if not member(i,jS) then j2 := i; break; end if;   
   end do;
   
   fv := simplify(subs(y=_v*x,e1));
   fvar := indets(fv,name);
   if member(x,fvar) or member(y,fvar) then
      error "the DE is not homogeneous"
   end if;
  
   intg := Int(1/(fv-_v),_v);
   if prntflg then
      print(``);
      print(`Homogeneous DE for which substituting   `,y=_v*x,`  gives . . `);
      print(simplify(Int(1/x,x))=simplify(intg));
      print(``);
   end if;
   val := value(intg);
   if prntflg then
      print(ln(x)=simplify(val)+C[j1]);print(``);
   end if;

   rightOK := indets(val,'specfunc(anything,int)')={};

   # Check whether value of integral is a sum of logs.
   if frmt='explicit' and rightOK then
      islg := islog(val,_v);
   else
      islg:=false;
   end if;
   
   if rightOK then
      rs := subs(_v=y/x,val);
   else
      rs := Int(1/(fv-_v),_v=``..y/x);
   end if;

   assignedconst := false;
   if islg then
      exprs := subs(_v=y/x,exp2(val,_v));
      soln := x=C[j2]*exprs;
      if initcond then
         const2 := traperror(eval(subs({x=x0,y=y0},
                             simplify(x/exprs))));
         if const2&lt;&gt;lasterror then
            soln := simplify(subs(C[j2]=const2,soln));
            assignedconst := true;
            lc := traperror(ln(const2));
            if lc&lt;&gt;lasterror then
               const := simplify(lc);
            end if;
         end if;
      end if;
   else
      soln := ln(x)=simplify(rs)+C[j1];
      if initcond then
         if rightOK then
            const := traperror(eval(subs({x=x0,y=y0},ln(x)-rs)));
            if const&lt;&gt;lasterror then
               soln := simplify(subs(C[j1]=const,soln));
               assignedconst := true;
            end if;
         else
            const := ln(x0)-Int(1/(fv-_v),_v=``..y0/x0);
            soln := ln(x)=int(1/(fv-_v),_v=y0/x0..y/x)+ln(x0);
            assignedconst := true;
         end if;
      end if;
   end if;

   soln2 := subs(y=yx,soln);

   # Try to obtain y explicitly.
   sols := [];
   if frmt='explicit' and rightOK then
      sols := goodsols([solve(soln,y)]);
   end if;
 
   if initcond then
      if nops(sols)=0 and not assignedconst then
         error "impossible initial condition"
      end if;
      # Check which solution fits the initial condition.
      if assignedconst then
         sol := [];    
         for i from 1 to nops(sols) do
            yy0 := traperror(simplify(value(subs(x=x0,sols[i]))));
            if yy0&lt;&gt;lasterror and signum(y0-yy0)=0 then
               sol := [simplify(sols[i])];
               break;
            end if;
         end do;
         sols := sol;
      else # Have another go at finding the constant.
         if islg then
            for i from 1 to nops(sols) do
               cc := traperror([solve(y0=subs(x=x0,sols[i]),C[j2])]);
               if cc=lasterror then
                  error "impossible initial condition"
               end if;
               if cc&lt;&gt;[] then
                  const2 := cc[1];
                  sols := [simplify(subs(C[j2]=const2,sols[i]))];
                  break;
               end if;
            end do;
            if cc=[] or has(cc,C) then
               error "impossible initial condition"
            end if;
         else
            for i from 1 to nops(sols) do
               cc := traperror([solve(y0=subs(x=x0,sols[i]),C[j1])]);
               if cc=lasterror then
                  error "impossible initial condition"
               end if;
               if cc&lt;&gt;[] then
                  const := cc[1];
                  sols := [simplify(subs(C[j1]=const,sols[i]))];
                  break;
               end if;
            end do;
            if cc=[]  or has(cc,C) then
               error "impossible initial condition"
            end if;
         end if;
      end if;
   end if;

   if prntflg then
      if islg then
         print(ln(x)=simplify(rs)+C[j1]);print(``);
         print(x=C[j2]*exprs);
         if initcond then
            print(`Applying the initial condition . .  `,
                                     C[j2]=simplify(const2));
         end if;
      else
         print(ln(x)=simplify(rs)+C[j1]);
         if initcond then
            print(`Applying the initial condition . .  `,
                                     C[j1]=simplify(const));
         end if;
      end if;
      print(``);
   end if;

   if nops(sols)&gt;0 then
      return seq(yx=simplify(sols[i]),i=1..nops(sols));
   else
      return soln2;
   end if;
end proc:</Text-field></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">Examples are given in the following sections.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="ParagraphStyle4"><Font style="_cstyle24">desolveHG</Font><Font style="_cstyle22">: general solution examples</Font></Text-field></Title><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 1</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  </Font><Equation input-equation="2*x^2;" style="2D Comment">NiMqJiIiIyIiIiokJSJ4R0YkRiU=</Equation><Font style="_cstyle27">   </Font><Equation input-equation="dy/dx = x^2+y^2;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJCUieEciIiNGJiokJSJ5R0YsRiY=</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := 2*x^2*diff(y(x),x)=x^2+y(x)^2;
desolveHG(de,y(x),info=true);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSNkZUc2Ii8sJComSSJ4R0YlIiIjLUklZGlmZkdJKnByb3RlY3RlZEdGLTYkLUkieUdGJTYjRilGKSIiIkYqLCYqJEYpRipGMiokRi9GKkYy</Equation></Text-field></Output><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiNJIUc2Ig==</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiVJSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HNiIvSSJ5R0YkKiZJI192R0YkIiIiSSJ4R0YkRilJLX5+Z2l2ZXN+Ln4ufkdGJA==</Equation></Text-field></Output><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMvLUkkSW50RzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2JCokSSJ4R0YpISIiRiwsJC1GJTYkKiQsKCIiIkYzKiRJI192R0YpIiIjRjNGNSEiI0YtRjVGNg==</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiNJIUc2Ig==</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMvLUkjbG5HNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYjSSJ4R0YpLCYqJCwmSSNfdkdGKSIiIiEiIkYwRjEhIiMmSSJDR0YpNiNGMEYw</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiNJIUc2Ig==</Equation></Text-field></Output><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMvLUkjbG5HNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYjSSJ4R0YpLCYqJkYrIiIiLCZJInlHRikhIiJGK0YuRjEiIiMmSSJDR0YpNiNGLkYu</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiNJIUc2Ig==</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMvLUkieUc2IjYjSSJ4R0YmKihGKCIiIiwoLUkjbG5HNiRJKnByb3RlY3RlZEdGL0koX3N5c2xpYkdGJkYnISIiIiIjRiomSSJDR0YmNiNGKkYqRiosJkYsRjFGM0YqRjE=</Equation></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de,y(x));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMvLUkieUc2IjYjSSJ4R0YmKihGKCIiIiwoISIjRiotSSNsbkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YmRidGKkkkX0MxR0YmRipGKiwmRi1GKkYyRiohIiI=</Equation></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 2</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">   </Font><Equation input-equation="x^2;" style="2D Comment">NiMqJCUieEciIiM=</Equation><Font style="_cstyle27">    </Font><Equation input-equation="dy/dx = y^2;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKiQlInlHIiIj</Equation><Font style="_cstyle27">   </Font></Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := x^2*diff(y(x),x) = y(x)^2;
desolveHG(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYpJSJ4RyIiIyIiIi0lJWRpZmZHNiQtJSJ5RzYjRihGKEYqKiQpRi5GKUYq</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKS1GJTYkKiZGKEYoKiYlI192R0YoLCZGL0YoRihGKkYoRipGLw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyUjX3ZHISIiLUYlNiMsJkYrIiIiRjBGLEYwJiUiQ0c2I0YwRjA=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyomJSJ5RyIiIkYnISIiRi4tRiU2IywkKiYsJkYsRi5GJ0YtRi1GJ0YuRi5GLSYlIkNHNiNGLUYt</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyoqJiUiQ0c2IyIiIyIiIiUieUchIiJGJEYqLCYqJkYrRipGJEYsRipGKkYsRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooJiUiQ0c2IyIiIyIiIkYnRi4sJkYnRi5GKiEiIkYwRjA=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">This differential equation can also be solved by separating the variables.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">desolveSV(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlRFRoZX5ERX5oYXN+c2VwYXJhYmxlfnZhcmlhYmxlc34ufi5+Rw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCokKSUieUciIiNGKCEiIkYrLUYlNiQqJkYoRigqJCklInhHRixGKEYtRjM=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLCQqJiIiIkYmJSJ5RyEiIkYoLCYqJkYmRiYlInhHRihGKCYlIkNHNiNGJkYm</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJComRiciIiIsJkYqISIiKiYmJSJDRzYjRipGKkYnRipGKkYsRiw=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqJkYnIiIiLCZGKUYpKiYlJF9DMUdGKUYnRilGKSEiIg==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 3</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27"> ( </Font><Equation input-equation="x^2-y^2;" style="2D Comment">NiMsJiokJSJ4RyIiIyIiIiokJSJ5R0YmISIi</Equation><Font style="_cstyle27">  ) </Font><Equation input-equation="dy/dx = 2*x*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKigiIiNGJiUieEdGJiUieUdGJg==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (x^2-y(x)^2)*diff(y(x),x) = 2*x*y(x);
desolveHG(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJiokKSUieEciIiMiIiJGLCokKS0lInlHNiNGKkYrRiwhIiJGLC0lJWRpZmZHNiRGL0YqRiwsJCooRitGLEYqRixGL0YsRiw=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkLUYlNiQqKCwmKiQpJSNfdkciIiNGKEYoRihGKkYoRjJGKiwmRihGKEYwRihGKkYyRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IywmIiIiRiwqJCklI192RyIiI0YsRiwhIiItRiU2I0YvRiwmJSJDRzYjRixGLA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyomLCYqJClGJyIiIyIiIkYwKiQpJSJ5R0YvRjBGMEYwRichIiMhIiItRiU2IyomRjNGMEYnRjVGMCYlIkNHNiNGMEYw</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyoqJiUiQ0c2IyIiIyIiIiwmRipGKiomRiQhIiMlInlHRilGKiEiIkYuRipGJEYv</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQvLSUieUc2IyUieEcsJiomIyIiIiIiI0YrJiUiQ0c2I0YsRitGKyomRipGKyokLCYqJClGLUYsRitGKyomIiIlRispRidGLEYrISIiRipGK0YrL0YkLCZGKUYrRjBGOA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQvLSUieUc2IyUieEcsJCooIiIjISIiJSRfQzFHRissJiIiIkYuKiQsJkYuRi4qKCIiJUYuKUYsRipGLilGJ0YqRi5GKyNGLkYqRi5GLkYuL0YkLCQqKEYqRitGLEYrLCZGLkYuRi9GK0YuRi4=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 4</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">( </Font><Equation input-equation="3*x-2*y;" style="2D Comment">NiMsJiomIiIkIiIiJSJ4R0YmRiYqJiIiI0YmJSJ5R0YmISIi</Equation><Font style="_cstyle27">   ) </Font><Equation input-equation="dy/dx = 4*x-3*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJiIiJUYmJSJ4R0YmRiYqJiIiJEYmJSJ5R0YmRig=</Equation><Font style="_cstyle27">   </Font></Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (3*x-2*y(x))*diff(y(x),x) = 4*x-3*y(x);
desolveHG(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJiomIiIkIiIiJSJ4R0YqRioqJiIiI0YqLSUieUc2I0YrRiohIiJGKi0lJWRpZmZHNiRGLkYrRiosJiomIiIlRipGK0YqRioqJkYpRipGLkYqRjE=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkKiYjRigiIiNGKC1GJTYkKiYsJiIiJEYqKiZGLkYoJSNfdkdGKEYoRigsKEYuRigqJkYzRihGNUYoRioqJClGNUYuRihGKEYqRjVGKEYq</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKy1GJTYjKiYsJiUjX3ZHRitGKyEiIkYrLCZGMUYrRixGMkYrRitGMiYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKy1GJTYjKigsJiomRixGK0YnRitGKyUieUchIiJGKywmRjJGM0YnRitGK0YnISIjRitGMyYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyomJiUiQ0c2IyIiIyIiIiwoRilGKiooIiIkRiolInlHRipGJCEiIkYvKiZGJCEiI0YuRilGKiNGL0Yp</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQvLSUieUc2IyUieEcsJiooIiIkIiIiIiIjISIiRidGK0YrKiYjRitGLEYrKiQsJiokKUYnRixGK0YrKiYiIiVGKykmJSJDRzYjRixGLEYrRitGL0YrRisvRiQsJkYpRitGLkYt</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQvLSUieUc2IyUieEcqJiwmKioiIiQiIiIiIiMhIiJGJ0YsJSRfQzFHRixGLComRi1GLiwmKiYpRi9GLUYsKUYnRi1GLEYsIiIlRiwjRixGLUYuRixGL0YuL0YkKiYsJkYqRixGMEYsRixGL0Yu</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 5</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  ( </Font><Equation input-equation="y+x;" style="2D Comment">NiMsJiUieUciIiIlInhHRiU=</Equation><Font style="_cstyle27">   ) </Font><Equation input-equation="dy/dx = y-x;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYlInlHRiYlInhHRig=</Equation><Font style="_cstyle27">   </Font></Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (y(x)+x)*diff(y(x),x) = y(x)-x;
desolveHG(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJi0lInlHNiMlInhHIiIiRitGLEYsLSUlZGlmZkc2JEYoRitGLCwmRihGLEYrISIi</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkLUYlNiQqJiwmJSNfdkdGKEYoRihGKCwmRihGKCokKUYwIiIjRihGKEYqRjBGKg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgqJiMiIiIiIiNGKy1GJTYjLCZGK0YrKiQpJSNfdkdGLEYrRitGKyEiIi0lJ2FyY3Rhbkc2I0YyRjMmJSJDRzYjRitGKw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgqJiMiIiIiIiNGKy1GJTYjKiYsJiokKUYnRixGK0YrKiQpJSJ5R0YsRitGK0YrRichIiNGKyEiIi0lJ2FyY3Rhbkc2IyomRjVGK0YnRjdGNyYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgqJiMiIiIiIiNGKy1GJTYjKiYsJiokKUYnRixGK0YrKiQpLSUieUdGJkYsRitGK0YrRichIiNGKyEiIi0lJ2FyY3Rhbkc2IyomRjVGK0YnRjhGOCYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLCoqJiMiIiIiIiNGJy0lI2xuRzYjKiYsJiokKSUieEdGKEYnRicqJCktJSJ5RzYjRjBGKEYnRidGJ0YwISIjRichIiItJSdhcmN0YW5HNiMqJkYzRidGMEY3RjctRipGNUY3JSRfQzFHRjciIiE=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 6</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">( </Font><Equation input-equation="1+x^2;" style="2D Comment">NiMsJiIiIkYkKiQlInhHIiIjRiQ=</Equation><Font style="_cstyle27">  ) </Font><Equation input-equation="dy/dx = x*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlInhHRiYlInlHRiY=</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := x^2*diff(y(x),x) = x^2+3*x*y(x)+y(x)^2;
desolveHG(de,info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYpJSJ4RyIiIyIiIi0lJWRpZmZHNiQtJSJ5RzYjRihGKEYqLCgqJEYnRipGKiooIiIkRipGKEYqRi5GKkYqKiQpRi5GKUYqRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKS1GJTYkKiZGKEYoLChGKEYoKiYiIiNGKCUjX3ZHRihGKCokKUYxRjBGKEYoRipGMQ==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiIiIkYqLCYlI192R0YqRipGKiEiIkYtJiUiQ0c2I0YqRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiwmJSJ5RyIiIkYnRiwhIiJGJ0YsRi0mJSJDRzYjRixGLA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooRiciIiIsKC0lI2xuR0YmRipGKkYqJiUiQ0c2I0YqISIiRiosJkYsRipGLkYxRjFGMQ==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooRiciIiIsKEYqRiotJSNsbkdGJkYqJSRfQzFHRipGKiwmRixGKkYuRiohIiJGMA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="ParagraphStyle4"><Font style="_cstyle24">desolveHG</Font><Font style="_cstyle22">: particular solution examples</Font></Text-field></Title><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 1</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  </Font><Equation input-equation="2*x^2;" style="2D Comment">NiMqJiIiIyIiIiokJSJ4R0YkRiU=</Equation><Font style="_cstyle27">   </Font><Equation input-equation="dy/dx = x^2+y^2;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJCUieEciIiNGJiokJSJ5R0YsRiY=</Equation><Font style="_cstyle27">  , </Font><Equation input-equation=" y(1) = 1/2" style="2D Comment">NiMvLSUieUc2IyIiIiomRidGJyIiIyEiIg==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := 2*x^2*diff(y(x),x)=x^2+y(x)^2;
ic := y(1)=1/2;
desolveHG({de,ic},info=true);
g := unapply(rhs(%),x);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLCQqKCIiIyIiIiklInhHRihGKS0lJWRpZmZHNiQtJSJ5RzYjRitGK0YpRiksJiokRipGKUYpKiQpRi9GKEYpRik=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiNGKSIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkKiYiIiNGKC1GJTYkKiZGKEYoLChGKEYoKiQpJSNfdkdGLUYoRigqJkYtRihGNEYoRipGKkY0RihGKA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiIiIyIiIiwmJSNfdkdGK0YrISIiRi5GLiYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqKCIiIyIiIkYnRissJiUieUchIiJGJ0YrRi5GKyYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiEiJQ==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqKEYnIiIiLCYtJSNsbkdGJkYpIiIjRilGKSwmRitGKSIiJUYpISIi</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCooOSQiIiIsJi0lI2xuRzYjRi1GLiIiI0YuRi4sJkYwRi4iIiVGLiEiIkYoRihGKA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqKEYnIiIiLCYtJSNsbkdGJkYpIiIjRilGKSwmRitGKSIiJUYpISIi</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot1:=DEtools[DEplot](de,y(x),x=1..10,y=0..7,arrows=medium,
                  color=COLOR(RGB,.5,0,1)):
plot2:=plot(g(x),x=1..10,y=0..7,color=coral,thickness=2):
plots[display]([plot1,plot2]);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="300" type="two-dimensional" 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layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Uebung 9.2a)</Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := x*diff(y(x),x)+y(x)=ln(x)+1;
ic := y(1)=3;
#desolveHG(de,info=true);
desolveHG({de,ic},y(x),info=true);
fakorisiert:=factor(%);
g := unapply(rhs(%),x);
#dsolve(de);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSNkZUc2Ii8sJiomSSJ4R0YlIiIiLUklZGlmZkdJKnByb3RlY3RlZEdGLTYkLUkieUdGJTYjRilGKUYqRipGL0YqLCYtSSNsbkc2JEYtSShfc3lzbGliR0YlRjFGKkYqRio=</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSNpY0c2Ii8tSSJ5R0YlNiMiIiIiIiQ=</Equation></Text-field><Text-field layout="Error" style="Error">Error, (in desolveHG) the DE is not homogeneous
</Text-field></Output><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSxmYWtvcmlzaWVydEc2Ii8tSSJ5R0YlNiMiIiIiIiQ=</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIiIkRiVGJUYl</Equation></Text-field></Output></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 2</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">   </Font><Equation input-equation="x^2;" style="2D Comment">NiMqJCUieEciIiM=</Equation><Font style="_cstyle27">    </Font><Equation input-equation="dy/dx = y^2;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKiQlInlHIiIj</Equation><Font style="_cstyle27">  , </Font><Equation input-equation="y(1)=2" style="2D Comment">NiMvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := x^2*diff(y(x),x) = y(x)^2;
ic := y(1)=2;
desolveHG({de,ic},info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYpJSJ4RyIiIyIiIi0lJWRpZmZHNiQtJSJ5RzYjRihGKEYqKiQpRi5GKUYq</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKS1GJTYkKiZGKEYoKiYlI192R0YoLCZGL0YoRihGKkYoRipGLw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyUjX3ZHISIiLUYlNiMsJkYrIiIiRjBGLEYwJiUiQ0c2I0YwRjA=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyomJSJ5RyIiIkYnISIiRi4tRiU2IywkKiYsJkYsRi5GJ0YtRi1GJ0YuRi5GLSYlIkNHNiNGLUYt</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyoqJiUiQ0c2IyIiIyIiIiUieUchIiJGJEYqLCYqJkYrRipGJEYsRipGKkYsRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiI0Yo</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooIiIjIiIiRidGKywmRidGK0YqISIiRi1GLQ==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">This differential equation can also be solved by separating the variables.</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">desolveSV({de,ic},info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlRFRoZX5ERX5oYXN+c2VwYXJhYmxlfnZhcmlhYmxlc34ufi5+Rw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCokKSUieUciIiNGKCEiIkYrLUYlNiQqJkYoRigqJCklInhHRixGKEYtRjM=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLCQqJiIiIkYmJSJ5RyEiIkYoLCYqJkYmRiYlInhHRihGKCYlIkNHNiNGJkYm</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiNGKCIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooIiIjIiIiRidGKywmRidGK0YqISIiRi1GLQ==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqJkYnIiIiLCZGKUYpKiYiIiMhIiJGJ0YpRi1GLQ==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Pruefung 2b)</Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de:=diff(y(x),x)-2*y(x)=cos(x)+x^2+1;
ic := y(0)=10^(-5)-23/20;
#desolveSV(de,info=true);
desolveHG({de,ic},y(x),info=true);
fakorisiert:=factor(%);
g := unapply(rhs(%),x);
#dsolve(de);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSNkZUc2Ii8sJi1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInlHRiU2I0kieEdGJUYvIiIiRiwhIiMsKC1JJGNvc0c2JEYqSShfc3lzbGliR0YlRi5GMCokRi8iIiNGMEYwRjA=</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSNpY0c2Ii8tSSJ5R0YlNiMiIiEjIScqKlw2IicrKzU=</Equation></Text-field></Output><Output><Text-field layout="Error" style="Error">Error, (in desolveHG) the DE is not homogeneous
</Text-field></Output><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSxmYWtvcmlzaWVydEc2Ii8tSSJ5R0YlNiMiIiEjIScqKlw2IicrKzU=</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIyEnKipcNiInKys1RiVGJUYl</Equation></Text-field></Output></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 3</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27"> ( </Font><Equation input-equation="x^2-y^2;" style="2D Comment">NiMsJiokJSJ4RyIiIyIiIiokJSJ5R0YmISIi</Equation><Font style="_cstyle27">  ) </Font><Equation input-equation="dy/dx = 2*x*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKigiIiNGJiUieEdGJiUieUdGJg==</Equation><Font style="_cstyle27">  , </Font><Equation input-equation="y(0)=1" style="2D Comment">NiMvLSUieUc2IyIiISIiIg==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (x^2-y(x)^2)*diff(y(x),x) = 2*x*y(x);
ic := y(0)=1;
desolveHG({de,ic},info=true);
g := unapply(rhs(%),x);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJiokKSUieEciIiMiIiJGLCokKS0lInlHNiNGKkYrRiwhIiJGLC0lJWRpZmZHNiRGL0YqRiwsJCooRitGLEYqRixGL0YsRiw=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiISIiIg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkLUYlNiQqKCwmKiQpJSNfdkciIiNGKEYoRihGKkYoRjJGKiwmRihGKEYwRihGKkYyRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IywmIiIiRiwqJCklI192RyIiI0YsRiwhIiItRiU2I0YvRiwmJSJDRzYjRixGLA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgtRiU2IyomLCYqJClGJyIiIyIiIkYwKiQpJSJ5R0YvRjBGMEYwRichIiMhIiItRiU2IyomRjNGMEYnRjVGMCYlIkNHNiNGMEYw</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyoqJiUiQ0c2IyIiIyIiIiwmRipGKiomRiQhIiMlInlHRilGKiEiIkYuRipGJEYv</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIyIiIg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiMiIiIiIiNGKiomRishIiIsJkYqRioqJiIiJUYqKUYnRitGKkYtRilGKg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwmIyIiIiIiI0YuKiZGLUYuKiQsJkYuRi4qJiIiJUYuKTkkRi9GLiEiIkYtRi5GLkYoRihGKA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiMiIiIiIiNGKiomRishIiIsJkYqRioqJiIiJUYqKUYnRitGKkYtRilGKg==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot1:=DEtools[DEplot](de,y(x),x=-0.1..0.6,y=0..1.1,arrows=medium,
                  dirgrid=[25,25],color=COLOR(RGB,.5,0,1)):
plot2:=plot(g(x),x=0..0.5,y=0..1.1,color=coral,thickness=2):
plots[display]([plot1,plot2]);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="300" type="two-dimensional" 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layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Pruefung 3a)</Text-field></Title><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de:=diff(y(x),x)-3*y(x)/sqrt(x)=0;
ic := y(0)=10^(-5)-23/20;
#desolveSV(de,info=true);
desolveHG({de,ic},y(x),info=true);
fakorisiert:=factor(%);
g := unapply(rhs(%),x);
#dsolve(de);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiooIiIkRi5GKkYuRi0jISIiIiIjRjJGMw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiISMhJyoqXDYiJysrNQ==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle21" style="_cstyle35">Error, (in desolveHG) the DE is not homogeneous
</Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSxmYWtvcmlzaWVydEcvLSUieUc2IyIiISMhJyoqXDYiJysrNQ==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSJnRyMhJyoqXDYiJysrNQ==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 4</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">( </Font><Equation input-equation="3*x-2*y;" style="2D Comment">NiMsJiomIiIkIiIiJSJ4R0YmRiYqJiIiI0YmJSJ5R0YmISIi</Equation><Font style="_cstyle27">   ) </Font><Equation input-equation="dy/dx = 4*x-3*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJiIiJUYmJSJ4R0YmRiYqJiIiJEYmJSJ5R0YmRig=</Equation><Font style="_cstyle27">  , </Font><Equation input-equation="y(0)=1" style="2D Comment">NiMvLSUieUc2IyIiISIiIg==</Equation><Font style="_cstyle27">    </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (3*x-2*y(x))*diff(y(x),x)=4*x-3*y(x);
ic := y(0)=1;
desolveHG({de,ic},info=true);
g := unapply(rhs(%),x);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJiomIiIkIiIiJSJ4R0YqRioqJiIiI0YqLSUieUc2I0YrRiohIiJGKi0lJWRpZmZHNiRGLkYrRiosJiomIiIlRipGK0YqRioqJkYpRipGLkYqRjE=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiISIiIg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkKiYjRigiIiNGKC1GJTYkKiYsJiIiJEYqKiZGLkYoJSNfdkdGKEYoRigsKEYuRigqJkYzRihGNUYoRioqJClGNUYuRihGKEYqRjVGKEYq</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKy1GJTYjKiYsJiUjX3ZHRitGKyEiIkYrLCZGMUYrRixGMkYrRitGMiYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKy1GJTYjKigsJiomRixGK0YnRitGKyUieUchIiJGKywmRjJGM0YnRitGK0YnISIjRitGMyYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyomJiUiQ0c2IyIiIyIiIiwoRilGKiooIiIkRiolInlHRipGJCEiIkYvKiZGJCEiI0YuRilGKiNGL0Yp</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIyIiIg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiooIiIkIiIiIiIjISIiRidGK0YrKiZGLEYtLCYqJClGJ0YsRitGKyIiJUYrI0YrRixGKw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwmKiYjIiIkIiIjIiIiOSRGMUYxKiYjRjFGMEYxKiQsJiokKUYyRjBGMUYxIiIlRjFGNEYxRjFGKEYoRig=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot(g(x),x=0..10,y=0..20);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="300" type="two-dimensional" width="400">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</Plot></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,y(0)=1},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiooIiIkIiIiIiIjISIiRidGK0YrKiZGLEYtLCYqJClGJ0YsRitGKyIiJUYrI0YrRixGKw==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 5</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  ( </Font><Equation input-equation="y+x;" style="2D Comment">NiMsJiUieUciIiIlInhHRiU=</Equation><Font style="_cstyle27">   ) </Font><Equation input-equation="dy/dx = y-x;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYlInlHRiYlInhHRig=</Equation><Font style="_cstyle27">  , </Font><Equation input-equation="y(1)=1" style="2D Comment">NiMvLSUieUc2IyIiIkYn</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := (y(x)+x)*diff(y(x),x) = y(x)-x;
ic := y(1)=1;
soln := desolveHG({de,ic},y(x),info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYsJi0lInlHNiMlInhHIiIiRitGLEYsLSUlZGlmZkc2JEYoRitGLCwmRihGLEYrISIi</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIkYp</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkLUYlNiQqJiwmJSNfdkdGKEYoRihGKCwmRihGKCokKUYwIiIjRihGKEYqRjBGKg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgqJiMiIiIiIiNGKy1GJTYjLCZGK0YrKiQpJSNfdkdGLEYrRitGKyEiIi0lJ2FyY3Rhbkc2I0YyRjMmJSJDRzYjRitGKw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCgqJiMiIiIiIiNGKy1GJTYjKiYsJiokKUYnRixGK0YrKiQpJSJ5R0YsRitGK0YrRichIiNGKyEiIi0lJ2FyY3Rhbkc2IyomRjVGK0YnRjdGNyYlIkNHNiNGK0Yr</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmKiYjRigiIiNGKC0lI2xuRzYjRixGKEYoKiYiIiUhIiIlI1BpR0YoRig=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSVzb2xuRy8tJSNsbkc2IyUieEcsKiomIyIiIiIiI0YtLUYnNiMqJiwmKiQpRilGLkYtRi0qJCktJSJ5R0YoRi5GLUYtRi1GKSEiI0YtISIiLSUnYXJjdGFuRzYjKiZGN0YtRilGOkY6KiZGLEYtLUYnNiNGLkYtRi0qJiIiJUY6JSNQaUdGLUYt</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEctJSdSb290T2ZHNiMsLComIiIjIiIiLSUjbG5HNiMqJiwmKiQpRidGLUYuRi4qJCklI19aR0YtRi5GLkYuRichIiNGLiEiIiomIiIlRi4tJSdhcmN0YW5HNiMqJkY4Ri5GJ0Y6Ri5GOiomRjxGLi1GMEYmRi5GOiomRi1GLi1GMDYjRi1GLkYuJSNQaUdGLg==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot1:=DEtools[DEplot](de,y(x),x=0..3,y=0..1.2,arrows=medium,
                  color=COLOR(RGB,.5,0,1)):
plot2:=plots[implicitplot](subs(y=y(x),soln),x=0..3,y=0..1,
                  color=coral,thickness=2,grid=[50,50]):
plots[display]([plot1,plot2]);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="300" type="two-dimensional" 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layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 6</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">( </Font><Equation input-equation="1+x^2;" style="2D Comment">NiMsJiIiIkYkKiQlInhHIiIjRiQ=</Equation><Font style="_cstyle27">  ) </Font><Equation input-equation="dy/dx = x*y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlInhHRiYlInlHRiY=</Equation><Font style="_cstyle27">  , </Font><Equation input-equation="y(1)=1" style="2D Comment">NiMvLSUieUc2IyIiIkYn</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := x^2*diff(y(x),x)=x^2+3*x*y(x)+y(x)^2;
ic := y(1)=1;
desolveHG({de,ic},info=true);
g := unapply(rhs(%),x);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvKiYpJSJ4RyIiIyIiIi0lJWRpZmZHNiQtJSJ5RzYjRihGKEYqLCgqJEYnRipGKiooIiIkRipGKEYqRi5GKkYqKiQpRi5GKUYqRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIkYp</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKS1GJTYkKiZGKEYoLChGKEYoKiYiIiNGKCUjX3ZHRihGKCokKUYxRjBGKEYoRipGMQ==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiIiIkYqLCYlI192R0YqRipGKiEiIkYtJiUiQ0c2I0YqRio=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJkYnIiIiLCYlInlHRipGJ0YqISIiRi0mJSJDRzYjRipGKg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiNGKCIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooRiciIiIsJiomIiIjRiotJSNsbkdGJkYqRipGKkYqRiosJkYsRipGKiEiIkYxRjE=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkKig5JCIiIiwmKiYiIiNGLy0lI2xuRzYjRi5GL0YvRi9GL0YvLCZGMUYvRi8hIiJGN0Y3RihGKEYo</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJCooRiciIiIsJiNGKiIiI0YqLSUjbG5HRiZGKkYqLCZGLkYqRiwhIiJGMUYx</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot1:=DEtools[DEplot](de,y(x),x=1..1.6,y=0..15,arrows=medium,
                  color=COLOR(RGB,.5,0,1)):
plot2:=plot(g(x),x=1..1.6,y=0..15,color=coral,thickness=2):
plots[display]([plot1,plot2]);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="300" type="two-dimensional" 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layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 7</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27"> </Font><Equation input-equation="dy/dx = sin(y/x),y(1) = 5;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiLSUkc2luRzYjKiYlInlHRiYlInhHRigvLUYtNiNGJiIiJg==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := diff(y(x),x)=sin(y(x)^2/x^2);
ic := y(1)=5;
soln := desolveHG({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwtJSRzaW5HNiMqJkYpIiIjRiwhIiM=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiIiJg==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSVzb2xuRy8tJSNsbkc2IyUieEctJSRpbnRHNiQqJiIiIkYuLCYtJSRzaW5HNiMqJCklI192RyIiI0YuRi5GNSEiIkY3L0Y1OyIiJiomLSUieUdGKEYuRilGNw==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">plot1:=DEtools[DEplot](de,y(x),x=1..4,y=4.6..5.8,arrows=medium,
                  dirgrid=[30,30],color=COLOR(RGB,.5,0,1)):
plot2:=plots[implicitplot](subs(y=y(x),soln),x=1..4,y=4.6..5.8,
                  color=coral,thickness=2,grid=[50,50]):
plots[display]([plot1,plot2]);</Text-field></Input><Output><Text-field layout="_pstyle20" style="_pstyle20"><Plot height="392" type="two-dimensional" 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layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">More examples</Text-field></Title><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 1</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">   </Font><Equation input-equation="dy/dx = x*y/(x^2+y^2),y(1) = 2;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKiglInhHRiYlInlHRiYsJiokRioiIiNGJiokRitGLkYmRigvLUYrNiNGJkYu</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := diff(y(x),x)=x*y(x)/(x^2+y(x)^2);
ic := y(1)=2;
desolveHG({de,ic},info=true);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqKEYsIiIiRilGLiwmKiQpRiwiIiNGLkYuKiQpRilGMkYuRi4hIiI=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiUlSUhvbW9nZW5lb3VzfkRFfmZvcn53aGljaH5zdWJzdGl0dXRpbmd+fn5HLyUieUcqJiUjX3ZHIiIiJSJ4R0YoJS1+fmdpdmVzfi5+Ln5H</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUkSW50RzYkKiYiIiJGKCUieEchIiJGKSwkLUYlNiQqJiwmRihGKCokKSUjX3ZHIiIjRihGKEYoRjIhIiRGMkYq</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKyomLCYqKEYsRistRiU2IyUjX3ZHRispRjJGLEYrRitGKyEiIkYrRjIhIiNGK0Y0JiUiQ0c2I0YrRis=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUjbG5HNiMlInhHLCYqJiMiIiIiIiNGKyomLCYqKEYsRistRiU2IyomJSJ5R0YrRichIiJGKylGM0YsRitGNCokKUYnRixGK0YrRitGMyEiI0YrRismJSJDRzYjRitGKw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmI0YoIiIpISIiLSUjbG5HNiMiIiNGKA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMlIUc=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqJi0lKUxhbWJlcnRXRzYjLCQqJiMiIiIiIiVGLyomLSUkZXhwRzYjRi5GLylGJyIiI0YvRi9GLyMhIiJGNkYnRi8=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcqJi0lKUxhbWJlcnRXRzYjLCQqJiMiIiIiIiVGLyomLSUkZXhwRzYjRi5GLylGJyIiI0YvRi9GLyMhIiJGNkYnRi8=</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 2</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">   </Font><Equation input-equation="dy/dx = 2*x*y/(x^2+y^2),y(1) = 2;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKioiIiNGJiUieEdGJiUieUdGJiwmKiRGK0YqRiYqJEYsRipGJkYoLy1GLDYjRiZGKg==</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := diff(y(x),x)=2*x*y(x)/(x^2+y(x)^2);
ic := y(1)=2;
desolveHG({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJCoqIiIjIiIiRixGMEYpRjAsJiokKUYsRi9GMEYwKiQpRilGL0YwRjAhIiJGMA==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiMiIiQiIiUiIiIqJkYrISIiLCYiIipGLComIiM7RiwpRiciIiNGLEYsI0YsRjRGLA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">dsolve({de,ic},y(x));
simplify(%);</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiMiIiQiIiUiIiIqKEYqRixGKyEiIiwmRixGLCooIiM7RiwiIipGLkYnIiIjRiwjRixGM0Ys</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvLSUieUc2IyUieEcsJiMiIiQiIiUiIiIqJkYrISIiLCYiIipGLComIiM7RiwpRiciIiNGLEYsI0YsRjRGLA==</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Example 3</Text-field></Title><Text-field layout="_pstyle16" style="ParagraphStyle5"><Font style="_cstyle27">  </Font><Equation input-equation="2*x^2;" style="2D Comment">NiMqJiIiIyIiIiokJSJ4R0YkRiU=</Equation><Font style="_cstyle27">   </Font><Equation input-equation="dy/dx = a*x*y/(x^2+y^2),y(1) = 2;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKiolImFHRiYlInhHRiYlInlHRiYsJiokRisiIiNGJiokRixGL0YmRigvLUYsNiNGJkYv</Equation><Font style="_cstyle27">   </Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">de := diff(y(x),x)=a*x*y(x)/(x^2+y(x)^2);
ic := y(1)=2;
desolveHG({de,ic},y(x));</Text-field></Input><Output><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqKiUiYUciIiJGLEYvRilGLywmKiQpRiwiIiNGL0YvKiQpRilGM0YvRi8hIiI=</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle34">  </Font></Text-field><Text-field layout="_pstyle19" style="ParagraphStyle6"><Equation style="2D Output">NiMvJSJ4RyoqKSIiIywkKiYiIiJGKiwmJSJhR0YqRiohIiJGLUYtRiopLCZGLEYtIiImRiosJCooRidGLUYsRipGK0YtRipGKikqJi0lInlHNiNGJEYqRiRGLUYpRiopLCQqJiwoKiZGLEYqKUYkRidGKkYqKiQpRjVGJ0YqRi0qJEY9RipGLUYqRiQhIiNGLSwkRjJGLUYq</Equation><Font style="_cstyle34">  </Font></Text-field></Output></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Text-field layout="_pstyle12" style="_pstyle12"/><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Tasks</Text-field></Title><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">In questions 1 to 9, solve the given initial value problem.</Text-field><Text-field layout="_pstyle12" style="_cstyle19">In each case plot the graph of the solution along with the associated direction field for the differential equation.</Text-field><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">You may need to use the </Font><Font style="_cstyle25">implicitplot</Font><Font style="_cstyle19"> function from the </Font><Font style="_cstyle25">plots</Font><Font style="_cstyle19"> package.</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle32">Note</Font><Font style="_cstyle19">: When setting up the differential equation to be solved by either </Font><Font style="_cstyle25">dsolve</Font><Font style="_cstyle19"> or </Font><Font style="_cstyle25">desolveHG</Font><Font style="_cstyle19">: </Font></Text-field><Text-field layout="_pstyle22" style="ParagraphStyle5"><Font style="_cstyle36">Use </Font><Font style="_cstyle37">diff(y(x),x)</Font><Font style="_cstyle36"> to replace the mathematical symbol  </Font><Equation input-equation="dy/dx" style="2D Comment">NiMqJiUjZHlHIiIiJSNkeEchIiI=</Equation><Font style="_cstyle36">  .</Font></Text-field><Text-field layout="_pstyle22" style="ParagraphStyle5"><Font style="_cstyle36">Replace y by y(x) </Font><Font style="_cstyle38">everywhere y appears in the differential equation</Font><Font style="_cstyle36">.</Font></Text-field><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q1 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">    </Font><Equation input-equation="x;" style="2D Comment">NiMlInhH</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dy/dx = x+y;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYlInhHRiYlInlHRiY=</Equation><Font style="_cstyle19">  ,   </Font><Equation input-equation="y(1) = 1" style="2D Comment">NiMvLSUieUc2IyIiIkYn</Equation><Font style="_cstyle19">   </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q2 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">    </Font><Equation input-equation="x*y^2;" style="2D Comment">NiMqJiUieEciIiIqJCUieUciIiNGJQ==</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dy/dx = y^3-x^3;" style="2D Comment">NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJCUieUciIiRGJiokJSJ4R0YsRig=</Equation><Font style="_cstyle19">  ,  </Font><Equation input-equation="y(1) = 2" style="2D Comment">NiMvLSUieUc2IyIiIiIiIw==</Equation><Font style="_cstyle19">    </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q3 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">     </Font><Equation input-equation="dy/dx = (y^2-3*x*y)/(y^2-x^2),y(1) = 1/10;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKiYsJiokJSJ5RyIiI0YmKigiIiRGJiUieEdGJkYsRiZGKEYmLCZGK0YmKiRGMEYtRihGKC8tRiw2I0YmKiZGJkYmIiM1Rig=</Equation><Font style="_cstyle19">    </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q4 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">     </Font><Equation input-equation="dy/dx = (x^2+y^2)/(x*y-5*x^2),y(1) = 0;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKiYsJiokJSJ4RyIiI0YmKiQlInlHRi1GJkYmLCYqJkYsRiZGL0YmRiYqJiIiJkYmRitGJkYoRigvLUYvNiNGJiIiIQ==</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q5 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">    </Font><Equation input-equation="``(x^4+y^4)" style="2D Comment">NiMtJSFHNiMsJiokJSJ4RyIiJSIiIiokJSJ5R0YpRio=</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dy/dx+2*x^3*y = 0,y(1) = 1;" style="2D Comment">NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRicqJCUieEciIiRGJyUieUdGJ0YnIiIhLy1GLzYjRidGJw==</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q6 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">    </Font><Equation input-equation="``(x^2+x*y+y^2);" style="2D Comment">NiMtJSFHNiMsKCokJSJ4RyIiIyIiIiomRihGKiUieUdGKkYqKiRGLEYpRio=</Equation><Font style="_cstyle19">   </Font><Equation input-equation="dy/dx+y^2 = 0,y(1) = 1;" style="2D Comment">NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyokJSJ5RyIiI0YnIiIhLy1GKzYjRidGJw==</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q7 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">     </Font><Equation input-equation="dy/dx = y/x+exp(y/x),y(1) = 1;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiLCYqJiUieUdGJiUieEdGKEYmLSUkZXhwRzYjRipGJi8tRis2I0YmRiY=</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section><Title><Text-field layout="_pstyle14" style="_cstyle22">Q8 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">     </Font><Equation input-equation="dy/dx = ``(y/x)*ln(y/x),y(1) = 1;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiKiYtJSFHNiMqJiUieUdGJiUieEdGKEYmLSUjbG5HRixGJi8tRi42I0YmRiY=</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Q9 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">     </Font><Equation input-equation="dy/dx = cosh(y/x)+y/x,y(1) = 0;" style="2D Comment">NiQvKiYlI2R5RyIiIiUjZHhHISIiLCYtJSVjb3NoRzYjKiYlInlHRiYlInhHRihGJkYtRiYvLUYuNiNGJiIiIQ==</Equation><Font style="_cstyle19">      </Font></Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="_pstyle14" style="_cstyle22">Q10 </Text-field></Title><Text-field layout="_pstyle12" style="ParagraphStyle5"><Font style="_cstyle19">(a) Construct a first order homogeneous differential equation, for which </Font><Font style="_cstyle25">dsolve</Font><Font style="_cstyle19"> or </Font><Font style="_cstyle25">desolveHG</Font><Font style="_cstyle19"> can find the general solution.</Font></Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">(b) For the differential equation chosen in (a) pick an initial condition and find the particular solution with this initial condition. </Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle12" style="_cstyle19">(c) Plot the graph of the particular solution along with the direction field associated with the differential equation.</Text-field><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_pstyle12"/><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_pstyle12"/></Input></Group><Text-field layout="_pstyle12" style="_cstyle19">_____________________________________</Text-field><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Group><Input><Text-field layout="_pstyle12" prompt="&gt; " style="_cstyle20">;</Text-field></Input></Group></Section><Text-field layout="_pstyle12" style="_pstyle12"/><Text-field layout="_pstyle23" style="_pstyle23"/><Text-field layout="_pstyle24" style="_pstyle24"/><Text-field/></Worksheet>