A procedure for solving 1st order exact DE's by Peter Stone, Dept. of Applied and Environmental Sciences, RMITpeter.stone@rmit.edu.au . . or . . peterstone@optusnet.com.auVersion: 10.12.2003restart;;

RMIT file path to read Maple m-file of differential equation solving procedures with the interface desolve. read "J:\\Class_Notes/Peter Stone/MapleMath/procdrs/DEsol.m";;Another file path. read "D:\\Maple 9/procedures/DEsol.m";;

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Calling Sequence: desolveEX( de ) desolveEX( {de,ic} ) desolveEX( {de,ic}, y(x) ) Parameters: de - a first order differential equation with the derivative given in the form diff(y(x),x), (if x and y are the independent and dependent variables respectively). ic - an initial condition in the form y(x0) = y0. Description:The procedure desolveEX attempts to solve an exact first order linear differential equation NiMsJi0lIk1HNiQlInhHJSJ5RyIiIi0lIk5HRiZGKQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh where NiMvLSUlRGlmZkc2JC0lIk1HNiQlInhHJSJ5R0YrLUYlNiQtJSJOR0YpRio= ,by constructing a function of two variables NiMtJSJGRzYkJSJ4RyUieUc= such that NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLSUiTUdGKQ== and NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLSUiTkdGKQ== ,by one of two methods:Method 1 (standard scheme) Let NiMvLSUiRkc2JCUieEclInlHLCYtJkYlNiMiIiJGJkYtLSUiZ0c2I0YoRi0= , where NiMvLSYlIkZHNiMiIiI2JCUieEclInlHLSUkSW50RzYkLSUiTUdGKUYq .Choose NiMtJSJnRzYjJSJ5Rw== so that NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLSUiTkdGKQ== by NiMvLSUiZ0c2IyUieUctJSRJbnRHNiQtJSFHNiMsJi0lIk5HNiQlInhHRiciIiItJSVkaWZmRzYkLSYlIkZHNiNGM0YxRichIiJGJw== .Method 2 (alternative scheme)Let NiMvLSUiRkc2JCUieEclInlHLCYtJkYlNiMiIiNGJiIiIi0lImhHNiNGJ0Yu , where NiMvLSYlIkZHNiMiIiM2JCUieEclInlHLSUkSW50RzYkLSUiTkdGKUYr .Choose NiMtJSJoRzYjJSJ4Rw== so that NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLSUiTUdGKQ== by NiMvLSUiaEc2IyUieEctJSRJbnRHNiQtJSFHNiMsJi0lIk1HNiRGJyUieUciIiItJSVkaWZmRzYkLSYlIkZHNiMiIiNGMUYnISIiRic= .In both cases the general solution has the form NiMvLSUiRkc2JCUieEclInlHJiUiQ0c2IyIiIg== . If an initial condition is provided, the value of the constant NiMmJSJDRzYjIiIi will be found.Options: info=true/false The option info=true causes intermediate steps in the solution to be printed.format=explicit/implicitWhen format=explicit, which is the default option, an attempt is made to solve for y explicitly.scheme=std/alt or scheme=standard/alternateWhen "scheme=std", or "scheme=standard", method 1 above is attempted, otherwise, if "scheme=alt", or "scheme=alternate", method 2 is attempted.The default is "scheme=std", but if this fails, method 2 will be attempted.

How to activate: To make the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter]. You can then close up the subsection.

desolveEX := proc(ff) local vars,derivs,x,yx,y,df,drv,prntflg,Options,lsde, j,intg,val,soln,initcond,de,ic,x0,y0,rsic,lsic,const, Mxy,Nxy,g,h,dMxy,dNxy,F1,gpy,gy,hpx,hx,F,Fxy,Rxy,la,N,M, goodsols,removeconsts,sols,frmt,assignedconst,i,schm, startopts,xx,yy,ee,cc,sol,yy0,tst,Cs,gt,jS,j1,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: # of goodsols # Remove any constants from a sum of terms removeconsts := proc(expr::algebraic) local terms,sum,i; if op(0,expr)<>`+`then return expr end if; terms := [op(expr)]; sum := 0; for i from 1 to nops(terms) do if not type(terms[i],complexcons) then sum := sum + terms[i]; end if; end do; sum; end proc: # of removeconsts # 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>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'; schm='std'; if nargs>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<>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 hasoption(Options,'scheme','schm','Options') then if not member(schm,{'std','standard','alt','alternate'}) then error "\"scheme\" must be 'standard','std','alternate' or 'alt'" end if; if schm='standard' then schm := 'std' elif schm='alternate' then schm := 'alt' end if; end if; if nops(Options)>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<>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<>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)<>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)<>x then error "the derivative, %1, does not make sense",df; end if; if startopts=3 then if x<>xx or y<>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; if rhs(de)=0 then lsde := lhs(de); else lsde := lhs(de)-rhs(de); end if; if patmatch(lsde,conditional(M::algebraic+ N::algebraic*diff(yx,x), indets(M,'specfunc(anything,diff)')={} and indets(N,'specfunc(anything,diff)')={}),'la') then Mxy := subs(yx=y,subs(la,M)); Nxy := subs(yx=y,subs(la,N)); elif patmatch(lsde,conditional(diff(yx,x)+R::algebraic, indets(R,'specfunc(anything,diff)')={}),'la') then Rxy := subs(yx=y,subs(la,R)); Mxy := numer(Rxy); Nxy := denom(Rxy); else drv := solve(de,df); if nops([drv])=1 then Rxy := subs(yx=y,drv); Mxy := -numer(Rxy); Nxy := denom(Rxy); else error "cannot extract required terms from the ODE" end if; end if; dMxy := diff(Mxy,y); dNxy := diff(Nxy,x); tst := testeq(dMxy=dNxy); if not tst or tst=FAIL then error "the DE is not exact" end if; if prntflg then print(``); print(`Test for exactness . .`); print(Diff(Mxy,y)=dMxy); print(Diff(Nxy,x)=dNxy); print(``); print(`Exact DE . . `,Mxy + (Nxy)*``(dy/dx)=0); print(``); 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; if schm=alt then goto(2222) end if; intg := Int(Mxy,x); if prntflg then print(`Let `,F(x,y)=simplify(intg)+g(y)); fi; val := value(intg); if indets(val,'specfunc(anything,int)')={} then F1 := simplify(combine(val,trig)); else if prntflg then print(`cannot find `,intg);print(``); end if; goto(2222); end if; if prntflg then print(` = `,F1+g(y));print(``); end if; gpy := simplify(combine(Nxy-diff(F1,y),trig)); if has(gpy,x) then if prntflg then print(gpy,`should not depend on`,x);print(``); end if; goto(2222); end if; if prntflg then print(`where `,Diff(g(y),y)=Nxy-Diff(F1,y)); print(` =`,gpy);print(``); end if; intg := Int(gpy,y); val := value(intg); if indets(val,'specfunc(anything,int)')={} then gy := simplify(combine(val,trig)); else if prntflg then print(`cannot find `,intg);print(``); end if; goto(2222); end if; if prntflg then print(`so `,g(y)=intg); print(` = `,gy);print(``); end if; Fxy := removeconsts(F1+gy); if prntflg then print(F(x,y)=Fxy,` satisfies . .`); print(Diff(F(x,y),x)=Mxy); print(Diff(F(x,y),y)=Nxy);print(``); end if; soln := Fxy=C[j1]; goto(1111); 2222: intg := Int(Nxy,y); if prntflg then print(`Let `,F(x,y)=simplify(intg)+h(x)); end if; val := value(intg); if indets(val,'specfunc(anything,int)')={} then F1 := simplify(combine(val,trig)); else if prntflg then print(`cannot find `,intg);print(``); end if; return NULL; end if; if prntflg then print(` = `,F1+h(x));print(``); end if; hpx := simplify(combine(Mxy-diff(F1,x),trig)); if has(hpx,y) then if prntflg then print(hpx,`should not depend on`,y);print(``); end if; return NULL; end if; if prntflg then print(`where `,Diff(h(x),x)=Mxy-Diff(F1,x)); print(` =`,hpx);print(``); end if; intg := Int(hpx,x); val := value(intg); if indets(val,'specfunc(anything,int)')={} then hx := simplify(combine(val,trig)); else if prntflg then print(`cannot find `,intg);print(``); end if; return NULL; end if; if prntflg then print(`so `,h(x)=intg); print(` = `,hx);print(``); end if; Fxy := removeconsts(F1+hx); if prntflg then print(F(x,y)=Fxy,` satisfies . .`); print(Diff(F(x,y),x)=Mxy); print(Diff(F(x,y),y)=Nxy);print(``); end if; soln := Fxy=C[j1]; 1111: if initcond then if prntflg then print(subs(y=yx,soln));print(``); end if; assignedconst := false; const := traperror(eval(subs({x=x0,y=y0},Fxy))); if const<>lasterror then soln := simplify(subs(C[j1]=const,soln)); assignedconst := true; end if; end if; # Try to obtain y explicitly. sols := []; if frmt='explicit' 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<>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. 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<>[] then const := cc[1]; sols := [simplify(subs(C[j1]=const,sols[i]))]; break; end if; end do; if cc=[] then error "impossible initial condition" end if; end if; if prntflg then print(`Applying the initial condition . . `, C[j1]=simplify(const));print(``); end if; end if; if nops(sols)>0 then return seq(yx=simplify(sols[i]),i=1..nops(sols)); else return subs(y=yx,soln); end if; end proc:;

Examples are given in the following sections.;

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NiMsKiIiIiEiIiomJSJ5R0YkLSUkZXhwRzYjKiYlInhHRiRGJ0YkRiRGJComRidGJC0lJGNvc0dGKkYkRiQtJSFHNiMsKEYkRiQqJkYsRiRGKEYkRiQqJkYsRiRGLkYkRiRGJA== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh M := (x,y)->-1+y*exp(x*y)+y*cos(x*y): N := (x,y)->1+x*exp(x*y)+x*cos(x*y): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: desolveEX(de,info=true);

NiNJIUc2Ig==NiNJN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5HNiI=NiMvLUklRGlmZkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQsKCEiIiIiIiomSSJ5R0YpRi0tSSRleHBHRiY2IyomSSJ4R0YpRi1GL0YtRi1GLSomRi9GLS1JJGNvc0dGJkYyRi1GLUYvLCpGMEYtKihGL0YtRjRGLUYwRi1GLUY2Ri0qKEYvRi0tSSRzaW5HRiZGMkYtRjRGLUYsNiMvLUklRGlmZkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQsKCIiIkYsKiZJInhHRilGLC1JJGV4cEdGJjYjKiZGLkYsSSJ5R0YpRixGLEYsKiZGLkYsLUkkY29zR0YmRjFGLEYsRi4sKkYvRiwqKEYzRixGLkYsRi9GLEYsRjVGLCooRjNGLC1JJHNpbkdGJkYxRixGLkYsISIiNiNJIUc2Ig==NiRJLkV4YWN0fkRFfi5+Ln5HNiIvLCohIiIiIiIqJkkieUdGJEYoLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiQ2IyomSSJ4R0YkRihGKkYoRihGKComRipGKC1JJGNvc0dGLUYwRihGKComLChGKEYoKiZGMkYoRitGKEYoKiZGMkYoRjRGKEYoRigtSSFHRiQ2IyomSSNkeUdGJEYoSSNkeEdGJEYnRihGKCIiIQ==NiNJIUc2Ig==

NiRJJUxldH5HNiIvLUkiRkdGJDYkSSJ4R0YkSSJ5R0YkLCYtSSRJbnRHNiRJKnByb3RlY3RlZEdGL0koX3N5c2xpYkdGJDYkLCghIiIiIiIqJkYqRjQtSSRleHBHRi42IyomRilGNEYqRjRGNEY0KiZGKkY0LUkkY29zR0YuRjhGNEY0RilGNC1JImdHRiQ2I0YqRjQ=NiRJJ35+PX5+fkc2IiwqSSJ4R0YkISIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiQ2IyomRiYiIiJJInlHRiRGL0YvLUkkc2luR0YqRi1GLy1JImdHRiQ2I0YwRi8=NiNJIUc2Ig==

NiRJJ3doZXJlfkc2Ii8tSSVEaWZmRzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiQ2JC1JImdHRiQ2I0kieUdGJEYvLCoiIiJGMSomSSJ4R0YkRjEtSSRleHBHRig2IyomRjNGMUYvRjFGMUYxKiZGM0YxLUkkY29zR0YoRjZGMUYxLUYnNiQsKEYzISIiRjRGMS1JJHNpbkdGKEY2RjFGL0Y+NiRJLH5+fn5+fn5+fn49RzYiIiIiNiNJIUc2Ig==NiRJJHNvfkc2Ii8tSSJnR0YkNiNJInlHRiQtSSRJbnRHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJDYkIiIiRik=NiRJJ35+PX5+fkc2IkkieUdGJA==NiNJIUc2Ig==NiQvLUkiRkc2IjYkSSJ4R0YmSSJ5R0YmLCpGKCEiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YmNiMqJkYoIiIiRilGM0YzLUkkc2luR0YuRjFGM0YpRjNJMH5+c2F0aXNmaWVzfi5+LkdGJg==NiMvLUklRGlmZkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQtSSJGR0YpNiRJInhHRilJInlHRilGLiwoISIiIiIiKiZGL0YyLUkkZXhwR0YmNiMqJkYuRjJGL0YyRjJGMiomRi9GMi1JJGNvc0dGJkY2RjJGMg==NiMvLUklRGlmZkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQtSSJGR0YpNiRJInhHRilJInlHRilGLywoIiIiRjEqJkYuRjEtSSRleHBHRiY2IyomRi5GMUYvRjFGMUYxKiZGLkYxLUkkY29zR0YmRjVGMUYxNiNJIUc2Ig==

NiMvLCpJInhHNiIhIiItSSRleHBHNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJjYjKiZGJSIiIi1JInlHRiY2I0YlRi9GLy1JJHNpbkdGKkYtRi9GMEYvJkkiQ0dGJjYjRi8=

dsolve(de,implicit);

NiMvLCwlInhHISIiLSUkZXhwRzYjKiZGJSIiIi0lInlHNiNGJUYrRistJSRzaW5HRilGK0YsRislJF9DMUdGKyIiIQ==

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NiMsKiIiIiEiIiomJSJ5R0YkLSUkZXhwRzYjKiYlInhHRiRGJ0YkRiRGJComRidGJC0lJGNvc0dGKkYkRiQtJSFHNiMsKEYkRiQqJkYsRiRGKEYkRiQqJkYsRiRGLkYkRiRGJA== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh M := (x,y)->-1+y*exp(x*y)+y*cos(x*y): N := (x,y)->1+x*exp(x*y)+x*cos(x*y): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: desolveEX(de,scheme=alt,info=true);

NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwoISIiIiIiKiYlInlHRiktJSRleHBHNiMqJiUieEdGKUYrRilGKUYpKiZGK0YpLSUkY29zR0YuRilGKUYrLCpGLEYpKihGK0YpRjBGKUYsRilGKUYyRikqKEYrRiktJSRzaW5HRi5GKUYwRilGKA== NiMvLSUlRGlmZkc2JCwoIiIiRigqJiUieEdGKC0lJGV4cEc2IyomRipGKCUieUdGKEYoRigqJkYqRigtJSRjb3NHRi1GKEYoRiosKkYrRigqKEYvRihGKkYoRitGKEYoRjFGKCooRi9GKC0lJHNpbkdGLUYoRipGKCEiIg== NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywqISIiIiIiKiYlInlHRictJSRleHBHNiMqJiUieEdGJ0YpRidGJ0YnKiZGKUYnLSUkY29zR0YsRidGJyomLChGJ0YnKiZGLkYnRipGJ0YnKiZGLkYnRjBGJ0YnRictJSFHNiMqJiUjZHlHRiclI2R4R0YmRidGJyIiIQ== NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywmLSUkSW50RzYkLCgiIiJGLyomRihGLy0lJGV4cEc2IyomRihGL0YpRi9GL0YvKiZGKEYvLSUkY29zR0YzRi9GL0YpRi8tJSJoRzYjRihGLw== NiQlJ35+PX5+fkcsKiUieUciIiItJSRleHBHNiMqJiUieEdGJkYlRiZGJi0lJHNpbkdGKUYmLSUiaEc2I0YrRiY= NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImhHNiMlInhHRissKiEiIiIiIiomJSJ5R0YuLSUkZXhwRzYjKiZGK0YuRjBGLkYuRi4qJkYwRi4tJSRjb3NHRjNGLkYuLUYmNiQsKEYwRi5GMUYuLSUkc2luR0YzRi5GK0Yt NiQlLH5+fn5+fn5+fn49RyEiIg== NiMlIUc= NiQlJHNvfkcvLSUiaEc2IyUieEctJSRJbnRHNiQhIiJGKA== NiQlJ35+PX5+fkcsJCUieEchIiI= NiMlIUc= NiQvLSUiRkc2JCUieEclInlHLCpGJyEiIi0lJGV4cEc2IyomRiciIiJGKEYvRi8tJSRzaW5HRi1GL0YoRi8lMH5+c2F0aXNmaWVzfi5+Lkc= NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCghIiIiIiIqJkYrRi4tJSRleHBHNiMqJkYqRi5GK0YuRi5GLiomRitGLi0lJGNvc0dGMkYuRi4= NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLCgiIiJGLSomRipGLS0lJGV4cEc2IyomRipGLUYrRi1GLUYtKiZGKkYtLSUkY29zR0YxRi1GLQ== NiMlIUc= NiMvLColInhHISIiLSUkZXhwRzYjKiZGJSIiIi0lInlHNiNGJUYrRistJSRzaW5HRilGK0YsRismJSJDRzYjRis=

dsolve(de,implicit);

NiMvLCwlInhHISIiLSUkZXhwRzYjKiZGJSIiIi0lInlHNiNGJUYrRistJSRzaW5HRilGK0YsRislJF9DMUdGKyIiIQ==

;

NiMsKComLSUkY29zRzYjJSJ4RyIiIi0lJHNpbkdGJ0YpRikqJkYoRikqJCUieUciIiNGKSEiIi0lIUc2IyomRi5GKSwmRilGKSokRihGL0YwRilGKQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh M := (x,y)->cos(x)*sin(x)-x*y^2: N := (x,y)->y*(1-x^2): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: desolveEX(de,info=true);

NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwmKiYtJSRjb3NHNiMlInhHIiIiLSUkc2luR0YrRi1GLSomRixGLSklInlHIiIjRi0hIiJGMiwkKiZGLEYtRjJGLSEiIw== NiMvLSUlRGlmZkc2JComJSJ5RyIiIiwmRilGKSokKSUieEciIiNGKSEiIkYpRi0sJComRi1GKUYoRikhIiM= NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywoKiYtJSRjb3NHNiMlInhHIiIiLSUkc2luR0YpRitGKyomRipGKyklInlHIiIjRishIiIqKEYwRissJkYrRisqJClGKkYxRitGMkYrLSUhRzYjKiYlI2R5R0YrJSNkeEdGMkYrRisiIiE= NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywmLSUkSW50RzYkLCYqJi0lJGNvc0c2I0YoIiIiLSUkc2luR0YyRjNGMyomRihGMylGKSIiI0YzISIiRihGMy0lImdHNiNGKUYz NiQlJ35+PX5+fkcsKi0lJGNvc0c2IywkJSJ4RyIiIyMhIiIiIiUjIiIiRi1GLComI0YvRipGLyomKUYpRipGLyklInlHRipGL0YvRiwtJSJnRzYjRjVGLw== NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImdHNiMlInlHRissJiomRisiIiIsJkYuRi4qJCklInhHIiIjRi4hIiJGLkYuLUYmNiQsKC0lJGNvc0c2IywkRjJGMyNGNCIiJSNGLkY9RjQqJiNGLkYzRi4qJkYxRi4pRitGM0YuRi5GNEYrRjQ= NiQlLH5+fn5+fn5+fn49RyUieUc= NiMlIUc= NiQlJHNvfkcvLSUiZ0c2IyUieUctJSRJbnRHNiRGKEYo NiQlJ35+PX5+fkcsJCokKSUieUciIiMiIiIjRilGKA== NiMlIUc= NiQvLSUiRkc2JCUieEclInlHLCgtJSRjb3NHNiMsJEYnIiIjIyEiIiIiJSomIyIiIkYuRjQqJilGJ0YuRjQpRihGLkY0RjRGMComRjNGNEY3RjRGNCUwfn5zYXRpc2ZpZXN+Ln4uRw== NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCYqJi0lJGNvc0c2I0YqIiIiLSUkc2luR0YwRjFGMSomRipGMSlGKyIiI0YxISIi NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrKiZGKyIiIiwmRi1GLSokKUYqIiIjRi0hIiJGLQ== NiMlIUc= NiQvLSUieUc2IyUieEcsJCooLCYhIiIiIiIqJClGJyIiI0YsRixGK0YvI0YsRi8sJComRipGLCwmLSUkY29zRzYjLCRGJ0YvRiwqJiIiJUYsJiUiQ0c2I0YsRixGLEYsRitGMEYwL0YkLCRGKSNGK0Yv

dsolve(de,implicit); dsolve(de);

NiMvLCYqJCktJSJ5RzYjJSJ4RyIiIyIiIkYsKiYsJiokKS0lJGNvc0dGKUYrRiwhIiIlJF9DMUdGLEYsLCZGM0YsKiQpRipGK0YsRixGM0YzIiIh NiQvLSUieUc2IyUieEcqJiwmISIiIiIiKiQpRiciIiNGK0YrRiosJComRilGKywmKiQpLSUkY29zR0YmRi5GK0YrJSRfQzFHRipGK0YqI0YrRi4vRiQsJEYoRio=

;

NiMsKComLSUkY29zRzYjJSJ4RyIiIi0lJHNpbkdGJ0YpRikqJkYoRikqJCUieUciIiNGKSEiIi0lIUc2IyomRi5GKSwmRilGKSokRihGL0YwRilGKQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh M := (x,y)->cos(x)*sin(x)-x*y^2: N := (x,y)->y*(1-x^2): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: desolveEX(de,scheme=alt,info=true);

NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwmKiYtJSRjb3NHNiMlInhHIiIiLSUkc2luR0YrRi1GLSomRixGLSklInlHIiIjRi0hIiJGMiwkKiZGLEYtRjJGLSEiIw== NiMvLSUlRGlmZkc2JComJSJ5RyIiIiwmRilGKSokKSUieEciIiNGKSEiIkYpRi0sJComRi1GKUYoRikhIiM= NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywoKiYtJSRjb3NHNiMlInhHIiIiLSUkc2luR0YpRitGKyomRipGKyklInlHIiIjRishIiIqKEYwRissJkYrRisqJClGKkYxRitGMkYrLSUhRzYjKiYlI2R5R0YrJSNkeEdGMkYrRisiIiE= NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywmKiYsJiEiIiIiIiokKUYoIiIjRi5GLkYuLSUkSW50RzYkRilGKUYuRi0tJSJoRzYjRihGLg== NiQlJ35+PX5+fkcsKCokKSUieUciIiMiIiIjRilGKComRipGKSomKSUieEdGKEYpRiZGKUYpISIiLSUiaEc2I0YuRik= NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImhHNiMlInhHRissKComLSUkY29zR0YqIiIiLSUkc2luR0YqRjBGMComRitGMCklInlHIiIjRjAhIiItRiY2JCwmKiRGNEYwI0YwRjYqJkY8RjAqJilGK0Y2RjBGNEYwRjBGN0YrRjc= NiQlLH5+fn5+fn5+fn49RywkLSUkc2luRzYjLCQlInhHIiIjIyIiIkYq NiMlIUc= NiQlJHNvfkcvLSUiaEc2IyUieEctJSRJbnRHNiQsJC0lJHNpbkc2IywkRigiIiMjIiIiRjFGKA== NiQlJ35+PX5+fkcsJC0lJGNvc0c2IywkJSJ4RyIiIyMhIiIiIiU= NiMlIUc= NiQvLSUiRkc2JCUieEclInlHLCgtJSRjb3NHNiMsJEYnIiIjIyEiIiIiJSomIyIiIkYuRjQqJilGJ0YuRjQpRihGLkY0RjRGMComRjNGNEY3RjRGNCUwfn5zYXRpc2ZpZXN+Ln4uRw== NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCYqJi0lJGNvc0c2I0YqIiIiLSUkc2luR0YwRjFGMSomRipGMSlGKyIiI0YxISIi NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrKiZGKyIiIiwmRi1GLSokKUYqIiIjRi0hIiJGLQ== NiMlIUc= NiQvLSUieUc2IyUieEcsJCooLCYhIiIiIiIqJClGJyIiI0YsRixGK0YvI0YsRi8sJComRipGLCwmLSUkY29zRzYjLCRGJ0YvRiwqJiIiJUYsJiUiQ0c2I0YsRixGLEYsRitGMEYwL0YkLCRGKSNGK0Yv

dsolve(de,implicit); dsolve(de);

NiMvLCYqJCktJSJ5RzYjJSJ4RyIiIyIiIkYsKiYsJiokKS0lJGNvc0dGKUYrRiwhIiIlJF9DMUdGLEYsLCZGM0YsKiQpRipGK0YsRixGM0YzIiIh NiQvLSUieUc2IyUieEcqJiwmISIiIiIiKiQpRiciIiNGK0YrRiosJComRilGKywmKiQpLSUkY29zR0YmRi5GK0YrJSRfQzFHRipGK0YqI0YrRi4vRiQsJEYoRio=

;

NiMsKComLSUkY29zRzYjJSJ4RyIiIi0lJHNpbkdGJ0YpRikqJkYoRikqJCUieUciIiNGKSEiIi0lIUc2IyomRi5GKSwmRilGKSokRihGL0YwRilGKQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh M := (x,y)->3*x^2*sin(y)^2: N := (x,y)->2*x^3*sin(y)*cos(y)-2*exp(2*y): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: desolveEX(de,info=true);

NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwkKiYpJSJ4RyIiIyIiIiktJSRzaW5HNiMlInlHRitGLCIiJEYxLCQqKEYpRixGLkYsLSUkY29zR0YwRiwiIic= NiMvLSUlRGlmZkc2JCwmKigpJSJ4RyIiJCIiIi0lJHNpbkc2IyUieUdGLC0lJGNvc0dGL0YsIiIjKiZGM0YsLSUkZXhwRzYjLCRGMEYzRiwhIiJGKiwkKigpRipGM0YsRi1GLEYxRiwiIic= NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywmKiYpJSJ4RyIiIyIiIiktJSRzaW5HNiMlInlHRilGKiIiJComLCYqKClGKEYwRipGLEYqLSUkY29zR0YuRipGKSomRilGKi0lJGV4cEc2IywkRi9GKUYqISIiRiotJSFHNiMqJiUjZHlHRiolI2R4R0Y8RipGKiIiIQ== NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywoLSUkSW50RzYkKiQpRigiIiMiIiJGKCIiJCooRjJGMUYrRjEpLSUkY29zRzYjRilGMEYxISIiLSUiZ0dGN0Yx NiQlJ35+PX5+fkcsKCokKSUieEciIiQiIiIjRikiIiMqJkYqRikqJkYmRiktJSRjb3NHNiMsJCUieUdGK0YpRikhIiItJSJnRzYjRjJGKQ== NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImdHNiMlInlHRissKCooKSUieEciIiQiIiItJSRzaW5HRipGMS0lJGNvc0dGKkYxIiIjKiZGNkYxLSUkZXhwRzYjLCRGK0Y2RjEhIiItRiY2JCwmKiRGLkYxI0YxRjYqJkZBRjEqJkYuRjEtRjVGOkYxRjFGPEYrRjw= NiQlLH5+fn5+fn5+fn49RywkLSUkZXhwRzYjLCQlInlHIiIjISIj NiMlIUc= NiQlJHNvfkcvLSUiZ0c2IyUieUctJSRJbnRHNiQsJC0lJGV4cEc2IywkRigiIiMhIiNGKA== NiQlJ35+PX5+fkcsJC0lJGV4cEc2IywkJSJ5RyIiIyEiIg== NiMlIUc= NiQvLSUiRkc2JCUieEclInlHLCgqJClGJyIiJCIiIiNGLSIiIyomRi5GLSomRitGLS0lJGNvc0c2IywkRihGL0YtRi0hIiItJSRleHBHRjRGNiUwfn5zYXRpc2ZpZXN+Ln4uRw== NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCQqJilGKiIiIyIiIiktJSRzaW5HNiNGK0YvRjAiIiQ= NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLCYqKClGKiIiJCIiIi0lJHNpbkc2I0YrRjAtJSRjb3NHRjNGMCIiIyomRjZGMC0lJGV4cEc2IywkRitGNkYwISIi NiMlIUc= NiMvLCgqJCklInhHIiIkIiIiI0YpIiIjKiZGKkYpKiZGJkYpLSUkY29zRzYjLCQtJSJ5RzYjRidGK0YpRikhIiItJSRleHBHRjBGNSYlIkNHNiNGKQ==

dsolve(de,implicit);

NiMvLCoqJCklInhHIiIkIiIiI0YpIiIjKiZGKkYpKiZGJkYpLSUkY29zRzYjLCQtJSJ5RzYjRidGK0YpRikhIiItJSRleHBHRjBGNSUkX0MxR0YpIiIh

;

;

;

We solve the initial value problem NiMsKiIiIiEiIiomJSJ5R0YkLSUkZXhwRzYjKiYlInhHRiRGJ0YkRiRGJComRidGJC0lJGNvc0dGKkYkRiQtJSFHNiMsKEYkRiQqJkYsRiRGKEYkRiQqJkYsRiRGLkYkRiRGJA== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh , NiMvLSUieUc2IyIiISomIiIlIiIiIiIkISIi , and plot the graph of the solution.M := (x,y)->-1+y*exp(x*y)+y*cos(x*y): N := (x,y)->1+x*exp(x*y)+x*cos(x*y): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: ic := y(0)=4/3; sol := desolveEX({de,ic},y(x));

NiM+JSNpY0cvLSUieUc2IyIiISMiIiUiIiQ= NiM+JSRzb2xHLywqJSJ4RyEiIi0lJGV4cEc2IyomRiciIiItJSJ5RzYjRidGLUYtLSUkc2luR0YrRi1GLkYtIyIiKCIiJA==

eq := subs(y(x)=y,sol); plots[implicitplot](eq,x=-4..4,y=-4..4,grid=[100,100], thickness=2,color=coral);

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;

We solve the initial value problem NiMsKComLSUkY29zRzYjJSJ4RyIiIi0lJHNpbkdGJ0YpRikqJkYoRikqJCUieUciIiNGKSEiIi0lIUc2IyomRi5GKSwmRilGKSokRihGL0YwRilGKQ== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh , y(0) = 1, and plot the graph of the solution for x between -1 and 1. M := (x,y)->cos(x)*sin(x)-x*y^2: N := (x,y)->y*(1-x^2): de := M(x,y(x))+N(x,y(x))*diff(y(x),x)=0: ic := y(0)=1; sol := desolveEX({de,ic},y(x));

NiM+JSNpY0cvLSUieUc2IyIiISIiIg== NiM+JSRzb2xHLy0lInlHNiMlInhHLCQqJiMiIiIiIiNGLSooLCZGLSEiIiokKUYpRi5GLUYtRjFGLkYsLCQqJkYwRi0sJi0lJGNvc0c2IywkKiZGLkYtRilGLUYtRi1GLUYtRi1GMUYsRi1GMQ==

g := unapply(rhs(sol),x); plot(g(x),x=-1..1,y=0..2,thickness=2,color=brown);

NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkKigsJiEiIiIiIiokKTkkIiIjRjBGMEYvRjQjRjBGNCwkKiZGLkYwLCYtJSRjb3NHNiMsJEYzRjRGMEYwRjBGMEYvRjUjRi9GNEYoRihGKA== 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

;

We solve the initial value problem NiMsJiooIiIjIiIiJSJ4R0YmJSJ5RyEiIkYmKiZGJ0YlKiRGKEYlRilGKQ== NiMvLSUhRzYjKiYlI2R5RyIiIiUjZHhHISIiIiIh , NiMvLSUieUc2IyIiIkYn . de := 2*x/y(x)-x^2/y(x)^2*diff(y(x),x)=0; ic := y(1)=1; desolveEX({de,ic},info=true);

NiM+JSNkZUcvLCYqJiUieEciIiItJSJ5RzYjRighIiIiIiMqKEYoRi5GKiEiIy0lJWRpZmZHNiRGKkYoRilGLSIiIQ== NiM+JSNpY0cvLSUieUc2IyIiIkYp NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwkKiYlInhHIiIiJSJ5RyEiIiIiI0YrLCQqJkYpRipGKyEiI0Yw NiMvLSUlRGlmZkc2JCwkKiYlInhHIiIjJSJ5RyEiIyEiIkYpLCQqJkYpIiIiRitGLEYs NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywmKiYlInhHIiIiJSJ5RyEiIiIiIyooRidGK0YpISIjLSUhRzYjKiYlI2R5R0YoJSNkeEdGKkYoRioiIiE= NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywmKiZGKSEiIi0lJEludEc2JEYoRigiIiIiIiMtJSJnRzYjRilGMA== NiQlJ35+PX5+fkcsJiomJSJ5RyEiIiUieEciIiMiIiItJSJnRzYjRiZGKg== NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImdHNiMlInlHRissJiomJSJ4RyIiI0YrISIjISIiLUYmNiQqJkYrRjFGLkYvRitGMQ== NiQlLH5+fn5+fn5+fn49RyIiIQ== NiMlIUc= NiQlJHNvfkcvLSUiZ0c2IyUieUctJSRJbnRHNiQiIiFGKA== NiQlJ35+PX5+fkciIiE= NiMlIUc= NiQvLSUiRkc2JCUieEclInlHKiZGKCEiIkYnIiIjJTB+fnNhdGlzZmllc34ufi5H NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCQqJkYqIiIiRishIiIiIiM= NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLCQqJkYqIiIjRishIiMhIiI= NiMlIUc= NiMvKiYtJSJ5RzYjJSJ4RyEiIkYoIiIjJiUiQ0c2IyIiIg== NiMlIUc= NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIkYo NiMlIUc= NiMvLSUieUc2IyUieEcqJClGJyIiIyIiIg==

;

We solve the initial value problem NiMsJiooIiIkIiIiKiQlInhHIiIjRiYqJCUieUdGKSEiIkYmKihGKUYmKiRGKEYlRiYqJEYrRiVGLEYs NiMvLSUhRzYjKiYlI2R5RyIiIiUjZHhHISIiLSUkc2luRzYjJSJ4Rw== , NiMvLSUieUc2IyIiIkYn ,and plot the graph of the solution. de := 3*x^2/y(x)^2-2*x^3/y(x)^3*diff(y(x),x)=sin(x); ic := y(1)=1; desolveEX({de,ic},info=true); g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYqJiUieEciIiMtJSJ5RzYjRighIiMiIiQqKkYpIiIiRihGLkYqISIkLSUlZGlmZkc2JEYqRihGMCEiIi0lJHNpbkdGLA== NiM+JSNpY0cvLSUieUc2IyIiIkYp NiMlIUc= NiMlN1Rlc3R+Zm9yfmV4YWN0bmVzc34ufi5H NiMvLSUlRGlmZkc2JCwmKiYlInhHIiIjJSJ5RyEiIyIiJC0lJHNpbkc2I0YpISIiRissJComRilGKkYrISIkISIn NiMvLSUlRGlmZkc2JCwkKiYlInhHIiIkJSJ5RyEiJCEiI0YpLCQqJkYpIiIjRitGLCEiJw== NiMlIUc= NiQlLkV4YWN0fkRFfi5+Ln5HLywoKiYlInhHIiIjJSJ5RyEiIyIiJC0lJHNpbkc2I0YnISIiKipGKCIiIkYnRitGKSEiJC0lIUc2IyomJSNkeUdGMSUjZHhHRi9GMUYvIiIh NiMlIUc= NiQlJUxldH5HLy0lIkZHNiQlInhHJSJ5RywmKiZGKSEiIy0lJEludEc2JCwmKiQpRigiIiMiIiIhIiQqJi0lJHNpbkc2I0YoRjQpRilGM0Y0RjRGKEY0ISIiLSUiZ0c2I0YpRjQ= NiQlJ35+PX5+fkcsJiomLCYqJCklInhHIiIkIiIiRisqJi0lJGNvc0c2I0YpRispJSJ5RyIiI0YrRitGK0YxISIjRistJSJnRzYjRjFGKw== NiMlIUc= NiQlJ3doZXJlfkcvLSUlRGlmZkc2JC0lImdHNiMlInlHRissJiomJSJ4RyIiJEYrISIkISIjLUYmNiQqJiwmKiQpRi5GLyIiIkY4KiYtJSRjb3NHNiNGLkY4KUYrIiIjRjhGOEY4RitGMUYrISIi NiQlLH5+fn5+fn5+fn49RyIiIQ== NiMlIUc= NiQlJHNvfkcvLSUiZ0c2IyUieUctJSRJbnRHNiQiIiFGKA== NiQlJ35+PX5+fkciIiE= NiMlIUc= NiQvLSUiRkc2JCUieEclInlHKiYsJiokKUYnIiIkIiIiRi4qJi0lJGNvc0c2I0YnRi4pRigiIiNGLkYuRi5GKCEiIyUwfn5zYXRpc2ZpZXN+Ln4uRw== NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YqLCYqJkYqIiIjRishIiMiIiQtJSRzaW5HNiNGKiEiIg== NiMvLSUlRGlmZkc2JC0lIkZHNiQlInhHJSJ5R0YrLCQqJkYqIiIkRishIiQhIiM= NiMlIUc= NiMvKiYsJiokKSUieEciIiQiIiJGKiomLSUkY29zRzYjRihGKiktJSJ5R0YuIiIjRipGKkYqRjAhIiMmJSJDRzYjRio= NiMlIUc= NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmRihGKC0lJGNvc0dGJ0Yo NiMlIUc= NiMvLSUieUc2IyUieEcsJCooLCgtJSRjb3NHRiYiIiJGLSEiIi1GLDYjRi1GLkYuLCQqJkYqRi1GJ0YtRi4jRi0iIiNGJ0YtRi4= NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkKigsKC0lJGNvc0c2IzkkIiIiRjMhIiItRjA2I0YzRjRGNCwkKiZGLkYzRjJGM0Y0I0YzIiIjRjJGM0Y0RihGKEYo

plot(g(x),x=0..15);

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

;

We solve the initial value problem NiMvLCYqJiUieEciIiMsJiomIiInIiIiJSJ5R0YrRitGK0YrRitGKyomLSUhRzYjLCYqJkYnRisqJEYmIiIkRitGKyomRjRGKyokRixGJ0YrRitGKy1GLzYjKiYlI2R5R0YrJSNkeEchIiJGK0YrIiIh , NiMvLSUieUc2IyIiISIiIg== , and plot the graph of the solution along with the gradient field. de := x^2*(6*y(x)+1)+(2*x^3+3*y(x)^2)*diff(y(x),x)=0; ic := y(0)=1; desolveEX({de,ic},info=true); g := unapply(rhs(%),x);

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plot1:=DEtools[DEplot](de,y(x),x=-1..3,y=-0.2..2, arrows=medium,dirgrid=[30,30],color=COLOR(RGB,.5,0,1)): plot2:=plot(g(x),x=-1..3,y=-0.2..2,color=coral, thickness=2): plots[display]([plot1,plot2]);

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G6$;$!2-++ILLL8"Fcieo$"2/++ILLL8$Fcieo;$!2/++ILLLt#F,$"2/++ILLL2#Fcieo-%*LINESTYLEG6#Fdido-%*GRIDSTYLEG6#%,RECTANGULARGF^jeo-F]ido6#%%NONEG-%,ORIENTATIONG6$$"#XFdidoF`\fo

;

;#Uebung 3.1.3a) de:=diff(y(x),x)-(2+x)*y(x)=0; bed := y(0)=exp(1); desolveEX({de,bed},info=true);

NiM+SSNkZUc2Ii8sJi1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInlHRiU2I0kieEdGJUYvIiIiKiYsJiIiI0YwRi9GMEYwRixGMCEiIiIiIQ==NiM+SSRiZWRHNiIvLUkieUdGJTYjIiIhLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2IyIiIg==Error, (in desolveEX) the DE is not exact

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Determine whether the differential equation NiMsKComIiIjIiIiJSJ4R0YmRiZGJiEiIi0lIUc2IywmKiYiIiRGJiUieUdGJkYmIiIoRiZGJg== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh is exact, and if so find its general solution.__________________________________________________________________________;

Determine whether the differential equation NiMsKComIiIlIiIiKiQlInhHIiIkRiZGJiooRiVGJkYoRiYlInlHRiZGJi0lIUc2IywoKiYiIiNGJiokRihGMUYmRiYqJkYxRiZGK0YmRiZGJiEiIkYm NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh is exact, and if so find its general solution._________________________________________________________________________________________;

Determine whether the differential equation NiMsKiokJSJ5RyIiJCIiIiomRiUiIiMtJSRzaW5HNiMlInhHRichIiJGLUYuLSUhRzYjLCYqKEYmRidGLUYnRiVGKUYnKihGKUYnRiVGJy0lJGNvc0dGLEYnRidGJw== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh is exact, and if so find its general solution.__________________________________________________________________________;

Determine whether the differential equation NiMsJiooJSJ5RyIiIiwmRiZGJiomIiIjRiYqJCUieEdGKUYmISIiRiYtJSRleHBHNiMsJkYqRiwqJEYlRilGLEYmRiYtJSFHNiMqKEYrRiYsJkYmRiYqJkYpRiZGMUYmRixGJkYtRiZGJg== NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh is exact, and if so, find its general solution.__________________________________________________________________________;

Solve the initial value problem NiMsJiokLCYlInhHIiIiJSJ5R0YnIiIjRictJSFHNiMsKCooRilGJ0YmRidGKEYnRicqJEYmRilGJ0YnISIiRic= NiQvKiYlI2R5RyIiIiUjZHhHISIiIiIhLy0lInlHNiNGJkYm and plot the graph of the solution (using implicitplot if necessary). __________________________________________________________________________;

Solve the initial value problem NiMsKC0lJGV4cEc2IyUieEciIiIlInlHRigtJSFHNiMsKCIiI0YoRidGKComRilGKC1GJTYjRilGKEYoRig= NiQvKiYlI2R5RyIiIiUjZHhHISIiIiIhLy0lInlHNiNGKUYm and plot the graph of the solution (using implicitplot if necessary). __________________________________________________________________________;

Solve the initial value problem NiMsKiomJSJ5RyIiIy0lJGNvc0c2IyUieEciIiJGKyooIiIkRisqJEYqRiZGK0YlRishIiIqJkYmRitGKkYrRi8tJSFHNiMsKCooRiZGK0YlRistJSRzaW5HRilGK0YrKiRGKkYtRi8tJSNsbkc2I0YlRitGKw== NiQvKiYlI2R5RyIiIiUjZHhHISIiIiIhLy0lInlHNiNGKS0lJGV4cEc2I0Ym and plot the graph of the solution (using implicitplot if necessary). __________________________________________________________________________;

Solve the initial value problem NiQvKiYsJiomIiM3IiIiKiQlInlHIiIjRihGKComJSJ4R0YoLSUkZXhwRzYjKiZGLUYoRipGKEYoISIiRigtJSFHNiMqJiUjZHlHRiglI2R4R0YyRigqJkYqRihGLkYoLy1GKjYjIiIhLCRGKEYy and plot the graph of the solution (using implicitplot if necessary). __________________________________________________________________________;

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