A procedure for solving 1st order linear DE's by Peter Stone, School of Life and Physical Sciences, RMITpeter.stone@rmit.edu.auVersion: 13.10.2004restart;;

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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;In order to solve a linear differential equation NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUicEc2IyUieEdGJyUieUdGJ0YnLSUicUdGLQ== ------- (i) first determine the integrating factor NiMvLSUickc2IyUieEctJSRleHBHNiMtJSRJbnRHNiQtJSJwR0YmRic=.Multiplying (i) by NiMtJSJ1RzYjJSJ4Rw== gives NiMtJSJyRzYjJSJ4Rw== NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooLSUickc2IyUieEdGJy0lInBHRi1GJyUieUdGJ0YnKiZGK0YnLSUicUdGLUYn or NiMvLCYqKC0lInJHNiMlInhHIiIiJSJ5R0YqLSUiJ0dGKEYqRioqKEYmRiotJSJwR0YoRiotRitGKEYqRioqJkYmRiotJSJxR0YoRio= ------- (ii).Now NiMvKiYlInJHIiIiLSUiJ0c2IyUieEdGJiomJSJkR0YmJSNkeEchIiI= NiM3Iy0lJGV4cEc2Iy0lJEludEc2JC0lInBHNiMlInhHRi0= NiMvJSFHLSUkZXhwRzYjLSUkSW50RzYkLSUicEc2IyUieEdGLg== NiMqJiUiZEciIiIlI2R4RyEiIg== NiM3Iy0lJEludEc2JC0lInBHNiMlInhHRio= NiMvJSFHKiYtJSRleHBHNiMtJSRJbnRHNiQtJSJwRzYjJSJ4R0YvIiIiRixGMA== NiMvJSFHKiYtJSJyRzYjJSJ4RyIiIi0lInBHRihGKg==, so (ii) can be written in the form NiMvLCYqKC0lInJHNiMlInhHIiIiJSJ5R0YqLSUiJ0dGKEYqRioqKEYnRipGLEYqLUYrRihGKkYqKiZGJkYqLSUicUdGKEYq ------- (iii).The left hand side of (iii) is the derivative of a product: NiMqJiUiZEciIiIlI2R4RyEiIg== NiM3IyomLSUickc2IyUieEciIiItJSJ5R0YnRik= so we have NiMqJiUiZEciIiIlI2R4RyEiIg== NiMvNyMqJi0lInlHNiMlInhHIiIiLSUickdGKEYqKiZGK0YqLSUicUdGKEYq,and NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lInJHRidGKS0lJEludEc2JComRipGKS0lInFHRidGKUYo, which gives NiMvLSUieUc2IyUieEcqJi0lJEludEc2JComLSUickdGJiIiIi0lInFHRiZGL0YnRi9GLSEiIg==, where NiMvLSUickc2IyUieEctJSRleHBHNiMtJSRJbnRHNiQtJSJwR0YmRic=. __________________________

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Calling Sequence: desolveLN( de ) desolveLN( {de,ic} ) desolveLN( {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 desolveLN attempts to solve a first order linear differential equation NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUicEc2IyUieEdGJyUieUdGJ0YnLSUicUdGLQ== by using the integrating factor NiMvLSUickc2IyUieEctJSRleHBHNiMtJSRJbnRHNiQtJSJwR0YmRic=, so that NiMvLSUieUc2IyUieEcqJiwmLSUkSW50RzYkKiYtJSJxR0YmIiIiLSUickdGJkYwRidGMCYlIkNHNiNGMEYwRjBGMSEiIg== .If no initial condition is given, a general solution with an arbitrary constant is sought.Options: info=true/false The option info=true causes intermediate steps in the solution to be printed.

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.

desolveLN := proc() local ff,vars,derivs,x,yx,y,df,dff,drv,e1,e2,prntflg,Options, j,sols,islog,opls,fx,gx,intg,intfact,val,soln,dy,dx, initcond,de,ic,x0,y0,rsic,lsic,f0,v0,const,fact, factOK,intOK,factgx,intg2,startopts,xx,yy,ee,d, Cs,gt,jS,j1,s,i,nvars; global C; if nargs>0 then ff := args[1] else error "at least one argument must be supplied" end if; initcond := false; if type(ff,{set(equation),list(equation)}) and nops(ff)=2 then ff := map(_u -> if has(_u,D) then convert(_u,diff) else _u end if,ff); 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 if has(de,'D') then de := convert(de,'diff') else de := ff end if; else error "the 1st argument, %1, is invalid .. it should be an equation or a set (or list) of 2 equations",ff; 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 "2nd argument, %1, has incorrect form for the dependent variable",ee; end if; end if; end if; prntflg := false; if nargs>=startopts 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 nops(Options)>0 then error "%1 is not a valid option for %2",op(1,Options),procname; end if; end if; derivs := indets(de,'specfunc(anything,diff)'); if derivs={} then error "the 1st argument, %1, is invalid .. it should be a differential equation or a set (or list) containing a differential equation and an initial condition",ff; 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 ODE 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 indicex j1 to use in the constant 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; # Handle the special case first if not member(y,indets(e1,name)) then if prntflg then print(``); print(`Linear DE . . `,Diff(yx,x)=e1); print(``); end if; intg := Int(e1,x); if prntflg then print(y=simplify(intg));print(``); end if; val := value(intg); intOK := indets(val,'specfunc(anything,int)')={}; if prntflg then print(y=simplify(val+C[j1]));print(``); end if; if initcond then if intOK then v0 := traperror(eval(subs(x=x0,val))); if v0=lasterror then error "impossible initial condition" end if; const := simplify(y0-v0); if prntflg then print(`Applying the initial condition . . `,C[j1]=const); print(``); end if; soln := yx=simplify(subs(C[j1]=const,val+C[j1])); else soln := yx=(Int(subs(x=_u,e1),_u=x0..x)+y0); end if; else if intOK then soln := yx=simplify(expand((val+C[j1]))); else soln := yx=Intat(subs(x=_u,e1),_u=x)+C[j1]; end if; end if; return soln; end if; # end of special case e2 := expand(e1); if not member(degree(e2,y),{0,1}) then error "the DE is not linear" end if; fx := -traperror(simplify(coeff(e2,y,1))); if fx=lasterror then error "the DE is not linear" end if; gx := combine(simplify(e2+fx*y),trig); if member(y,indets(gx,name)) then error "the DE is not linear" end if; if prntflg then print(``); print(`Linear DE . . `,Diff(yx,x)+fx*yx=gx); print(``); end if; intg := Int(fx,x); if prntflg then print(`Integrating factor . . `,exp(intg)); end if; val := value(intg); factOK := indets(val,'specfunc(anything,int)')={}; if factOK then intfact := simplify(exp(val)); if prntflg then print(` = `,intfact);print(``); end if; factgx := simplify(gx*intfact); intg := Int(factgx,x); if prntflg then print(yx*intfact=intg);print(``); end if; val := value(intg); intOK := indets(val,'specfunc(anything,int)')={}; if intOK then intg2 := val; else intg2 := Intat(subs(x=_u,factgx),_u=x); end if; else intfact := exp(Intat(subs(x=_u,fx),_u=x)); intg2 := Intat(subs(x=_v,gx)* exp(Intat(subs(x=_u,fx),_u=_v)),_v=x); end if; if prntflg then print(yx*intfact=intg2+C[j1]);print(``); end if; if initcond then if indets(intfact,'specfunc(anything,int)')={} then f0 := traperror(eval(subs(x=x0,intfact))); if f0=lasterror or f0=0 then error "impossible initial condition" end if; else f0 := exp(Intat(fx,x=x0)); end if; if indets(val,'specfunc(anything,int)')={} then v0 := traperror(eval(subs(x=x0,val))); if v0=lasterror then error "impossible initial condition" end if; const := simplify(y0*f0-v0); if prntflg then print(`Applying the initial condition . . `,C[j1]=const); print(``); end if; fact := simplify(1/intfact); soln := yx=simplify(fact*intg2)+const*fact; else const := y0*f0-Intat(subs(x=_v,intfact*gx),_v=x0); if prntflg then print(`Applying the initial condition . . `,C[j1]=const); print(``); end if; soln := yx=(Int(subs(x=_v,intfact*gx),_v=x0..x)+y0*f0)/intfact; end if; else fact := simplify(1/intfact); soln := yx=simplify(fact*intg2)+C[j1]*fact; end if; return soln; end proc:;

Examples are given in the following sections.;

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NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomIiIkRiclInlHRidGJy0lJGV4cEc2IyomIiIjRiclInhHRic= de := diff(y(x),x)+3*y(x)=exp(2*x); desolveLN(de,y(x),info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomIiIkRi5GKkYuRi4tJSRleHBHNiMsJComIiIjRi5GLUYuRi4=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomIiIkRi1GKUYtRi0tJSRleHBHNiMsJComIiIjRi1GLEYtRi0=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiQlInhHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkKiYiIiQiIiIlInhHRipGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIkIiIiJSJ4R0YrRitGKy0lInlHNiNGLEYrLSUkSW50RzYkLUYmNiMsJComIiImRitGLEYrRitGLA==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIkIiIiJSJ4R0YrRitGKy0lInlHNiNGLEYrLCYqJiNGKyIiJkYrLUYmNiMsJComRjNGK0YsRitGK0YrRismJSJDRzYjRitGKw==NiMlIUc=NiMvLSUieUc2IyUieEcsJiomIyIiIiIiJkYrLSUkZXhwRzYjLCQqJiIiI0YrRidGK0YrRitGKyomJiUiQ0c2I0YrRistRi42IywkKiYiIiRGK0YnRishIiJGK0Yr

dsolve(de,y(x));

NiMvLSUieUc2IyUieEcqJiwmLSUkZXhwRzYjLCRGJyIiJiMiIiJGLiUkX0MxR0YwRjAtRis2IywkRichIiRGMA==

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NiMlInhH NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyUieUdGKSokJSJ4RyIiJQ== de := x*diff(y(x),x)-y(x)=x^4; desolveLN(de,info=true);

NiM+JSNkZUcvLCYqJiUieEciIiItJSVkaWZmRzYkLSUieUc2I0YoRihGKUYpRi0hIiIqJClGKCIiJUYpNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomRiwhIiJGKUYtRi8qJClGLCIiJEYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQsJComIiIiRiwlInhHISIiRi5GLQ==NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUlInhHISIiNiMlIUc=NiMvKiYlInhHISIiLSUieUc2I0YlIiIiLSUkSW50RzYkKiQpRiUiIiNGKkYlNiMlIUc=NiMvKiYlInhHISIiLSUieUc2I0YlIiIiLCYqJiIiJEYmRiVGLUYqJiUiQ0c2I0YqRio=NiMlIUc=NiMvLSUieUc2IyUieEcsJiomIiIkISIiRiciIiUiIiIqJiYlIkNHNiNGLUYtRidGLUYt

dsolve(de,y(x));

NiMvLSUieUc2IyUieEcqJiwmKiQpRiciIiQiIiIjRi1GLCUkX0MxR0YtRi1GJ0Yt

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NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRiclInhHRiclInlHRidGJyomLSUkZXhwRzYjLCQqJiIiJEYnRixGJ0YpRicsJiomRitGJ0YsRidGJ0Y0RilGJw== de := diff(y(x),x)+2*x*y(x)=exp(-3*x)*(2*x-3); desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiooIiIjRi5GLUYuRipGLkYuKiYtJSRleHBHNiMsJComIiIkRi5GLUYuISIiRi4sJiomRjBGLkYtRi5GLkY3RjhGLg==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiooIiIjRi1GLEYtRilGLUYtLCYqKEYvRi0tJSRleHBHNiMsJComIiIkRi1GLEYtISIiRi1GLEYtRi0qJkY3Ri1GMkYtRjg=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQsJComIiIjIiIiJSJ4R0YtRi1GLg==NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IyokKSUieEciIiMiIiI=NiMlIUc=NiMvKiYtJSRleHBHNiMqJCklInhHIiIjIiIiRiwtJSJ5RzYjRipGLC0lJEludEc2JComLUYmNiMsJiomIiIkRixGKkYsISIiRihGLEYsLCYqJkYrRixGKkYsRixGOEY5RixGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMqJCklInhHIiIjIiIiRiwtJSJ5RzYjRipGLCwmLUYmNiMsJiomIiIkRixGKkYsISIiRihGLEYsJiUiQ0c2I0YsRiw=NiMlIUc=NiMvLSUieUc2IyUieEcsJi0lJGV4cEc2IywkKiYiIiQiIiJGJ0YvISIiRi8qJiYlIkNHNiNGL0YvLUYqNiMsJCokKUYnIiIjRi9GMEYvRi8=

dsolve(de,y(x));

NiMvLSUieUc2IyUieEcqJiwmLSUkZXhwRzYjKiZGJyIiIiwmISIkRi5GJ0YuRi5GLiUkX0MxR0YuRi4tRis2IywkKiQpRiciIiNGLiEiIkYu

;

NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUkdGFuRzYjJSJ4R0YnJSJ5R0YnRictJSRzaW5HNiMqJiIiI0YnRi5GJw== de := diff(y(x),x)+tan(x)*y(x)=sin(2*x); desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUkdGFuR0YsRi5GKkYuRi4tJSRzaW5HNiMsJComIiIjRi5GLUYuRi4=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUkdGFuR0YrRi1GKUYtRi0tJSRzaW5HNiMsJComIiIjRi1GLEYtRi0=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSR0YW5HNiMlInhHRi0=NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUtJSRjb3NHNiMlInhHISIiNiMlIUc=NiMvKiYtJSRjb3NHNiMlInhHISIiLSUieUdGJyIiIi0lJEludEc2JComLSUkc2luRzYjLCQqJiIiI0YsRihGLEYsRixGJUYpRig=NiMlIUc=NiMvKiYtJSRjb3NHNiMlInhHISIiLSUieUdGJyIiIiwmKiYiIiNGLEYlRixGKSYlIkNHNiNGLEYsNiMlIUc=NiMvLSUieUc2IyUieEcsJiomIiIjIiIiKS0lJGNvc0dGJkYqRishIiIqJiYlIkNHNiNGK0YrRi1GK0Yr

dsolve(de,y(x));

NiMvLSUieUc2IyUieEcsJiomIiIjIiIiKS0lJGNvc0dGJkYqRishIiIqJkYtRislJF9DMUdGK0Yr

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NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUkdGFuRzYjJSJ4R0YnJSJ5R0YnRicqJEYuIiIk de := diff(y(x),x)+tan(x)*y(x)=x^3; desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUkdGFuR0YsRi5GKkYuRi4qJClGLSIiJEYuNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUkdGFuR0YrRi1GKUYtRi0qJClGLCIiJEYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSR0YW5HNiMlInhHRi0=NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUtJSRjb3NHNiMlInhHISIiNiMlIUc=NiMvKiYtJSRjb3NHNiMlInhHISIiLSUieUdGJyIiIi0lJEludEc2JComRigiIiRGJUYpRig=NiMlIUc=NiMvKiYtJSRjb3NHNiMlInhHISIiLSUieUdGJyIiIiwmLSUkaW50RzYkKiYlI191RyIiJC1GJjYjRjJGKS9GMjslIUdGKEYsJiUiQ0c2I0YsRiw=NiMlIUc=NiMvLSUieUc2IyUieEcsJiomLSUkY29zR0YmIiIiLSUkaW50RzYkKiYlI191RyIiJC1GKzYjRjEhIiIvRjE7JSFHRidGLEYsKiYmJSJDRzYjRixGLEYqRixGLA==

dsolve(de,y(x));

NiMvLSUieUc2IyUieEcsJiomLSUkY29zR0YmIiIiLSUkSW50RzYkKiZGJyIiJEYqISIiRidGLEYsKiZGKkYsJSRfQzFHRixGLA==

subs(%,de); simplify(%);

NiMvLCYtJSVkaWZmRzYkLCYqJi0lJGNvc0c2IyUieEciIiItJSRJbnRHNiQqJkYtIiIkRiohIiJGLUYuRi4qJkYqRi4lJF9DMUdGLkYuRi1GLiomLSUkdGFuR0YsRi5GKEYuRi4qJClGLUYzRi4=NiMvKiQpJSJ4RyIiJCIiIkYk

;

NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomIiIjRiclInhHRidGJyIiJg== de := diff(x(t),t)+2*x(t)=5; desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieEc2IyUidEdGLSIiIiomIiIjRi5GKkYuRi4iIiY=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieEc2IyUidEdGLCIiIiomIiIjRi1GKUYtRi0iIiY=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiMlInRHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkKiYiIiMiIiIlInRHRipGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIjIiIiJSJ0R0YrRitGKy0lInhHNiNGLEYrLSUkSW50RzYkLCQqJiIiJkYrRiVGK0YrRiw=NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIjIiIiJSJ0R0YrRitGKy0lInhHNiNGLEYrLCYqJiMiIiZGKkYrRiVGK0YrJiUiQ0c2I0YrRis=NiMlIUc=NiMvLSUieEc2IyUidEcsJiMiIiYiIiMiIiIqJiYlIkNHNiNGLEYsLSUkZXhwRzYjLCQqJkYrRixGJ0YsISIiRixGLA==

dsolve(de);

NiMvLSUieEc2IyUidEcsJiMiIiYiIiMiIiIqJi0lJGV4cEc2IywkRichIiNGLCUkX0MxR0YsRiw=

;

NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomIiIjRiclInhHRidGJyomLSUkZXhwRzYjLCQqJkYrRiclInRHRidGKUYnLSUkc2luRzYjRjNGJw== de := diff(x(t),t)+2*x(t)=exp(-2*t)*sin(t); desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieEc2IyUidEdGLSIiIiomIiIjRi5GKkYuRi4qJi0lJGV4cEc2IywkKiZGMEYuRi1GLiEiIkYuLSUkc2luR0YsRi4=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieEc2IyUidEdGLCIiIiomIiIjRi1GKUYtRi0qJi0lJGV4cEc2IywkKiZGL0YtRixGLSEiIkYtLSUkc2luR0YrRi0=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiMlInRHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkKiYiIiMiIiIlInRHRipGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIjIiIiJSJ0R0YrRitGKy0lInhHNiNGLEYrLSUkSW50RzYkLSUkc2luR0YvRiw=NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIjIiIiJSJ0R0YrRitGKy0lInhHNiNGLEYrLCYtJSRjb3NHRi8hIiImJSJDRzYjRitGKw==NiMlIUc=NiMvLSUieEc2IyUidEcsJiomLSUkZXhwRzYjLCQqJiIiIyIiIkYnRjAhIiJGMC0lJGNvc0dGJkYwRjEqJiYlIkNHNiNGMEYwRipGMEYw

dsolve(de,x(t));

NiMvLSUieEc2IyUidEcsJiomLSUkZXhwRzYjLCRGJyEiIyIiIi0lJGNvc0dGJkYvISIiKiZGKkYvJSRfQzFHRi9GLw==

;

NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomIiIjRiclInhHRidGJy0lJHNpbkc2IyUidEc= de := diff(x(t),t)+2*x(t)=sin(t); desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieEc2IyUidEdGLSIiIiomIiIjRi5GKkYuRi4tJSRzaW5HRiw=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieEc2IyUidEdGLCIiIiomIiIjRi1GKUYtRi0tJSRzaW5HRis=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiMlInRHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkJSJ0RyIiIw==NiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKS0lJEludEc2JComLSUkc2luR0YnRilGKkYpRig=NiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKSwoKiYtJSRjb3NHRidGKUYqRikjISIiIiImKigjRi5GNUYpLSUkc2luR0YnRilGKkYpRikmJSJDRzYjRilGKQ==NiMlIUc=NiMvLSUieEc2IyUidEcsKC0lJGNvc0dGJiMhIiIiIiYqJiMiIiNGLSIiIi0lJHNpbkdGJkYxRjEqJiYlIkNHNiNGMUYxLSUkZXhwRzYjLCRGJyEiI0YxRjE=

dsolve(de,x(t));

NiMvLSUieEc2IyUidEcsKC0lJGNvc0dGJiMhIiIiIiYqJiMiIiNGLSIiIi0lJHNpbkdGJkYxRjEqJi0lJGV4cEc2IywkRichIiNGMSUkX0MxR0YxRjE=

subs(%,de); simplify(%);

NiMvLCYtJSVkaWZmRzYkLCYqJi0lJGNvc0c2IyUieEciIiItJSRJbnRHNiQqJiokKUYtIiIkRi5GLkYqISIiRi1GLkYuKiZGKkYuJSRfQzFHRi5GLkYtRi4qJi0lJHRhbkdGLEYuRihGLkYuRjM=NiMvKiQpJSJ4RyIiJCIiIkYk

;

NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomIiIjRiclInhHRidGJywqLSUkc2luRzYjJSJ0R0YnKiYtRi82IyomIiIkRidGMUYnRidGNkYpRicqJi1GLzYjKiYiIiZGJ0YxRidGJ0Y7RilGJyomLUYvNiMqJiIiKEYnRjFGJ0YnRkBGKUYn de := diff(x(t),t)+2*x(t)=sin(t)+sin(3*t)/3+sin(5*t)/5+sin(7*t)/7; desolveLN(de,info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieEc2IyUidEdGLSIiIiomIiIjRi5GKkYuRi4sKi0lJHNpbkdGLEYuKiYjRi4iIiRGLi1GMzYjLCRGLUY2Ri5GLiomI0YuIiImRi4tRjM2IywkRi1GPEYuRi4qJiNGLiIiKEYuLUYzNiMsJEYtRkJGLkYuNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieEc2IyUidEdGLCIiIiomIiIjRi1GKUYtRi0sKi0lJHNpbkdGK0YtKiYjRi0iIiRGLS1GMjYjLCRGLEY1Ri1GLSomI0YtIiImRi0tRjI2IywkRixGO0YtRi0qJiNGLSIiKEYtLUYyNiMsJEYsRkFGLUYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiMlInRHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkJSJ0RyIiIw==NiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKS0lJEludEc2JCwkKiYsKi0lJHNpbkdGJyIkMCIqJiIjTkYpLUY2NiMsJEYoIiIkRilGKSomIiNARiktRjY2IywkRigiIiZGKUYpKiYiIzpGKS1GNjYjLCRGKCIiKEYpRilGKUYqRikjRilGN0YoNiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKSw0KiYtJSRjb3NHRidGKUYqRikjISIiIiImKigjRi5GNUYpRipGKS0lJHNpbkdGJ0YpRikqJiNGKSIjOEYpKiYtRjI2IywkRigiIiRGKUYqRilGKUY0KigjRi4iI1JGKUYqRiktRjlGP0YpRikqJiNGKSIjSEYpKiZGKkYpLUYyNiMsJEYoRjVGKUYpRjQqKCNGLiIkWCJGKUYqRiktRjlGS0YpRikqJiNGKSIjYEYpKiZGKkYpLUYyNiMsJEYoIiIoRilGKUY0KigjRi4iJHIkRilGKkYpLUY5RlZGKUYpJiUiQ0c2I0YpRik=NiMlIUc=NiMvLSUieEc2IyUidEcsNC0lJGNvc0dGJiMhIiIiIiYqJiMiIiNGLSIiIi0lJHNpbkdGJkYxRjEqJiNGMSIjOEYxLUYqNiMsJEYnIiIkRjFGLComI0YwIiNSRjEtRjNGOEYxRjEqJiNGMSIjSEYxLUYqNiMsJEYnRi1GMUYsKiYjRjAiJFgiRjEtRjNGQ0YxRjEqJiNGMSIjYEYxLUYqNiMsJEYnIiIoRjFGLComI0YwIiRyJEYxLUYzRk1GMUYxKiYmJSJDRzYjRjFGMS0lJGV4cEc2IywkRichIiNGMUYx

;

NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUiZkc2IyUieEdGJyUieUdGJ0YnLSUiZ0dGLQ== f := 'f': g := 'g': de := diff(y(x),x)+f(x)*y(x)=g(x); desolveLN(de,info=true); factor(%);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUiZkdGLEYuRipGLkYuLSUiZ0dGLA==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUiZkdGK0YtRilGLUYtLSUiZ0dGKw==NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSJmRzYjJSJ4R0YtNiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEc2Iy0lJGludEc2JC0lImZHNiMlI191Ry9GMzslIUdGKEYpLCYtRi42JComLSUiZ0c2IyUjX3ZHRiktRis2Iy1GLjYkRjAvRjM7RjZGPkYpL0Y+RjVGKSYlIkNHNiNGKUYpNiMlIUc=NiMvLSUieUc2IyUieEcsJiomLSUkZXhwRzYjLCQtJSRpbnRHNiQtJSJmRzYjJSNfdUcvRjQ7JSFHRichIiIiIiItRi82JComLSUiZ0c2IyUjX3ZHRjktRis2Iy1GLzYkRjEvRjQ7RjdGQEY5L0ZARjZGOUY5KiYmJSJDRzYjRjlGOUYqRjlGOQ==NiMvLSUieUc2IyUieEcqJi0lJGV4cEc2IywkLSUkaW50RzYkLSUiZkc2IyUjX3VHL0YzOyUhR0YnISIiIiIiLCYtRi42JComLSUiZ0c2IyUjX3ZHRjgtRio2Iy1GLjYkRjAvRjM7RjZGQEY4L0ZARjVGOCYlIkNHNiNGOEY4Rjg=

dsolve(de);

NiMvLSUieUc2IyUieEcqJiwmLSUkSW50RzYkKiYtJSJnR0YmIiIiLSUkZXhwRzYjLUYrNiQtJSJmR0YmRidGMEYnRjAlJF9DMUdGMEYwLUYyNiMtRis2JCwkRjYhIiJGJ0Yw

;

;

;

#de := diff(y(x),x)-x*diff(y(x),x)-y(x)+(x-1)^2=0; de := diff(y(x),x)-x*diff(y(x),x)-y(x)=-(x-1)^2; ic := y(2)=0; desolveLN({de,ic},info=true);

NiM+SSNkZUc2Ii8sKC1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInlHRiU2I0kieEdGJUYvIiIiKiZGL0YwRihGMCEiIkYsRjIsJCokLCZGL0YwRjJGMCIiI0YyNiM+SSNpY0c2Ii8tSSJ5R0YlNiMiIiMiIiE=

NiNJIUc2Ig==NiRJMExpbmVhcn5ERX4ufi5+fkc2Ii8sJi1JJURpZmZHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJDYkLUkieUdGJDYjSSJ4R0YkRjAiIiIqJiwmRjBGMSEiIkYxRjRGLUYxRjFGMw==NiNJIUc2Ig==NiRJOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkc2Ii1JJGV4cEc2JEkqcHJvdGVjdGVkR0YoSShfc3lzbGliR0YkNiMtSSRJbnRHRic2JCokLCZJInhHRiQiIiIhIiJGMUYyRjA=NiRJNX5+fn5+fn5+fn5+fn5+fn49fn5+RzYiLCZJInhHRiQiIiIhIiJGJw==NiNJIUc2Ig==NiMvKiYtSSJ5RzYiNiNJInhHRiciIiIsJkYpRiohIiJGKkYqLUkkSW50RzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRic2JCokRisiIiNGKQ==NiNJIUc2Ig==NiMvKiYtSSJ5RzYiNiNJInhHRiciIiIsJkYpRiohIiJGKkYqLCYqJEYrIiIkI0YqRi8mSSJDR0YnNiNGKkYqNiNJIUc2Ig==NiRJRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkc2Ii8mSSJDR0YkNiMiIiIjISIiIiIkNiNJIUc2Ig==NiMvLUkieUc2IjYjSSJ4R0YmLCYqJCwmRigiIiIhIiJGLCIiIyNGLCIiJCokRitGLSNGLUYw

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomIiIkRiclInlHRidGJy0lJGV4cEc2IyomIiIjRiclInhHRicvLUYsNiMiIiFGMQ== de := diff(y(x),x)+3*y(x)=exp(2*x); ic := y(0)=2; desolveLN({de,ic},info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomIiIkRi5GKkYuRi4tJSRleHBHNiMsJComIiIjRi5GLUYuRi4=NiM+JSNpY0cvLSUieUc2IyIiISIiIw==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomIiIkRi1GKUYtRi0tJSRleHBHNiMsJComIiIjRi1GLEYtRi0=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiQlInhHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkKiYiIiQiIiIlInhHRipGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIkIiIiJSJ4R0YrRitGKy0lInlHNiNGLEYrLSUkSW50RzYkLUYmNiMsJComIiImRitGLEYrRitGLA==NiMlIUc=NiMvKiYtJSRleHBHNiMsJComIiIkIiIiJSJ4R0YrRitGKy0lInlHNiNGLEYrLCYqJiNGKyIiJkYrLUYmNiMsJComRjNGK0YsRitGK0YrRismJSJDRzYjRitGKw==NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiMiIioiIiY=NiMlIUc=NiMvLSUieUc2IyUieEcsJiomIyIiIiIiJkYrLSUkZXhwRzYjLCQqJiIiI0YrRidGK0YrRitGKyomIyIiKkYsRistRi42IywkKiYiIiRGK0YnRishIiJGK0Yr

dsolve({de,ic},y(x));

NiMvLSUieUc2IyUieEcqJiwmKiYjIiIiIiImRiwtJSRleHBHNiMsJComRi1GLEYnRixGLEYsRiwjIiIqRi1GLEYsLUYvNiMsJComIiIkRixGJ0YsISIiRiw=

;

NiMlInhH NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyUieUdGKSokJSJ4RyIiJS8tRio2I0YnIiIk de := x*diff(y(x),x)-y(x)=x^4; ic := y(1)=3; desolveLN({de,ic},info=true);

NiM+JSNkZUcvLCYqJiUieEciIiItJSVkaWZmRzYkLSUieUc2I0YoRihGKUYpRi0hIiIqJClGKCIiJUYpNiM+JSNpY0cvLSUieUc2IyIiIiIiJA==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomRiwhIiJGKUYtRi8qJClGLCIiJEYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQsJComIiIiRiwlInhHISIiRi5GLQ==NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUlInhHISIiNiMlIUc=NiMvKiYlInhHISIiLSUieUc2I0YlIiIiLSUkSW50RzYkKiQpRiUiIiNGKkYlNiMlIUc=NiMvKiYlInhHISIiLSUieUc2I0YlIiIiLCYqJiIiJEYmRiVGLUYqJiUiQ0c2I0YqRio=NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiMiIikiIiQ=NiMlIUc=NiMvLSUieUc2IyUieEcsJiomIyIiIiIiJEYrKiQpRiciIiVGK0YrRisqJiMiIilGLEYrRidGK0Yr

dsolve({de,ic},y(x));

NiMvLSUieUc2IyUieEcqJiwmKiQpRiciIiQiIiIjRi1GLCMiIilGLEYtRi1GJ0Yt

;

#Uebung 3.2.4a) de:=diff(y(x),x)+y(x)/x = (y(x))^3; #odeadvisor(de,[separable]); desolveLN(de,y(x),info=true);

NiM+SSNkZUc2Ii8sJi1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInlHRiU2I0kieEdGJUYvIiIiKiZGLEYwRi8hIiJGMCokRiwiIiQ=Error, (in desolveLN) the DE is not linear

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRiclInhHRiclInlHRidGJyomLSUkZXhwRzYjLCQqJiIiJEYnRixGJ0YpRicsJiomRitGJ0YsRidGJ0Y0RilGJy8tRi02IyIiIUYr de := diff(y(x),x)+2*x*y(x)=exp(-3*x)*(2*x-3); ic := y(0)=2; desolveLN({de,ic},info=true): f := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiooIiIjRi5GLUYuRipGLkYuKiYtJSRleHBHNiMsJComIiIkRi5GLUYuISIiRi4sJiomRjBGLkYtRi5GLkY3RjhGLg==NiM+JSNpY0cvLSUieUc2IyIiISIiIw==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiooIiIjRi1GLEYtRilGLUYtLCYqKEYvRi0tJSRleHBHNiMsJComIiIkRi1GLEYtISIiRi1GLEYtRi0qJkY3Ri1GMkYtRjg=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQsJComIiIjIiIiJSJ4R0YtRi1GLg==NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IyokKSUieEciIiMiIiI=NiMlIUc=NiMvKiYtJSRleHBHNiMqJCklInhHIiIjIiIiRiwtJSJ5RzYjRipGLC0lJEludEc2JComLUYmNiMsJiomIiIkRixGKkYsISIiRihGLEYsLCYqJkYrRixGKkYsRixGOEY5RixGKg==NiMlIUc=NiMvKiYtJSRleHBHNiMqJCklInhHIiIjIiIiRiwtJSJ5RzYjRipGLCwmLUYmNiMsJiomIiIkRixGKkYsISIiRihGLEYsJiUiQ0c2I0YsRiw=NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIkYoNiMlIUc=NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwmLSUkZXhwRzYjLCQqJiIiJCIiIjkkRjMhIiJGMy1GLjYjLCQqJClGNCIiI0YzRjVGM0YoRihGKA==

plot(f(x),x=0..2);

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

;

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUkdGFuRzYjJSJ4R0YnJSJ5R0YnRictJSRzaW5HNiMqJiIiI0YnRi5GJy8tRi82IyIiIUYn de := diff(y(x),x)+tan(x)*y(x)=sin(2*x); ic := y(0)=1; desolveLN({de,ic},info=true): g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUkdGFuR0YsRi5GKkYuRi4tJSRzaW5HNiMsJEYtIiIjNiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUkdGFuR0YrRi1GKUYtRi0tJSRzaW5HNiMsJEYsIiIjNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSR0YW5HNiMlInhHRi0=NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUtJSRjb3NHNiMlInhHISIiNiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGNvc0dGJyEiIi0lJEludEc2JComLSUkc2luRzYjLCRGKCIiI0YpRipGLEYoNiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGNvc0dGJyEiIiwmRiohIiMmJSJDRzYjRilGKQ==NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiIiJA==NiMlIUc=NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwmKiQpLSUkY29zRzYjOSQiIiMiIiIhIiMqJiIiJEY0Ri9GNEY0RihGKEYo

plot(g(x),x=0..Pi/2);

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

;

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUkdGFuRzYjJSJ4R0YnJSJ5R0YnRicqJEYuIiIkLy1GLzYjIiIhRic= de := diff(y(x),x)+tan(x)*y(x)=x^3; ic := y(0)=1; desolveLN({de,ic},info=true): g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUkdGFuR0YsRi5GKkYuRi4qJClGLSIiJEYuNiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUkdGFuR0YrRi1GKUYtRi0qJClGLCIiJEYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSR0YW5HNiMlInhHRi0=NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyomIiIiRiUtJSRjb3NHNiMlInhHISIiNiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGNvc0dGJyEiIi0lJEludEc2JComRigiIiRGKkYsRig=NiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGNvc0dGJyEiIiwmLSUkaW50RzYkKiYlI191RyIiJC1GKzYjRjJGLC9GMjslIUdGKEYpJiUiQ0c2I0YpRik=NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmRihGKC0lJEludEc2JComJSNfdkciIiQtJSRjb3NHNiNGLiEiIi9GLjslIUciIiFGMw==NiMlIUc=NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKComLCYtJSRJbnRHNiQqJiUjX3ZHIiIkLSUkY29zRzYjRjIhIiIvRjI7IiIhOSQiIiJGPEY8RjwtRjU2I0Y7RjxGKEYoRig=

plot(g(x),x=0..Pi/2);

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

;

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRiclInlHRiclInhHRilGJyomRidGJywmRi1GJy0lJGV4cEc2I0YtRidGKS8tRiw2I0YnRis= de := diff(y(x),x)+2*y(x)/x=1/(x+exp(x)); ic := y(1)=2; desolveLN({de,ic},info=true); g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiooIiIjRi5GKkYuRi0hIiJGLiomRi5GLiwmRi1GLi0lJGV4cEdGLEYuRjE=NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiooIiIjRi1GKUYtRiwhIiJGLSomRi1GLSwmRixGLS0lJGV4cEdGK0YtRjA=NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQsJComIiIiRiwlInhHISIiIiIjRi0=NiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+RyokKSUieEciIiMiIiI=NiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIilGKCIiI0YpLSUkSW50RzYkKiYsJkYoRiktJSRleHBHRidGKSEiIkYoRitGKA==NiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIilGKCIiI0YpLCYtJSRpbnRHNiQqJiwmJSNfdUdGKS0lJGV4cEc2I0YyRikhIiJGMkYrL0YyOyUhR0YoRikmJSJDRzYjRilGKQ==NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmIiIjRigtJSRJbnRHNiQqJiwmJSNfdkdGKC0lJGV4cEc2I0YwRighIiJGMEYqL0YwOyUhR0YoRjQ=NiMlIUc=NiMvLSUieUc2IyUieEcqJiwmLSUkSW50RzYkKiYsJiUjX3ZHIiIiLSUkZXhwRzYjRi9GMCEiIkYvIiIjL0YvO0YwRidGMEY1RjBGMEYnISIjNiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKComLCYtJSRJbnRHNiQqJiwmJSNfdkciIiItJSRleHBHNiNGM0Y0ISIiRjMiIiMvRjM7RjQ5JEY0RjlGNEY0RjwhIiNGKEYoRig=

Plotting the solution along with the gradient field suggests that the solution is correct.with(DEtools):with(plots): plot1:=DEplot(de,y(x),x=1..3,y=0..3,arrows=medium, color=blue): plot2:=plot(g(x),x=1..3,y=0..3,color=coral,thickness=2): display([plot1,plot2]);

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As a second check we plot a numerical solution together with the analytical solution. ff := dsolve({de,ic},y(x),type=numeric,output=listprocedure); gn := subs(ff,y(x)): plot3 := plot('gn'(x),x=1..3,y=0..3,color=black): display([plot3,plot2],labels=[`x`,`y(x)`]);

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

Maple 's dsolve gives the same result.de := diff(y(x),x) + 2*y(x)/x = 1/(x+exp(x)); ic := y(1)=2; dsolve({de,ic},y(x)): h := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiooIiIjRi5GKkYuRi0hIiJGLiomRi5GLiwmRi1GLi0lJGV4cEdGLEYuRjE=NiM+JSNpY0cvLSUieUc2IyIiIiIiIw==NiM+JSJoR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKComLCYtJSRJbnRHNiQqJiwmJSJ1RyIiIi0lJGV4cEc2I0YzRjQhIiJGMyIiIy9GMztGNDkkRjRGOUY0RjRGPCEiI0YoRihGKA==

;

Given NiMvLSUiZkc2IyUieEctJSpQSUVDRVdJU0VHNiQ3JCIiIjFGJ0YsNyQiIiEyRixGJw==, we find a continuous solution which satisfies the differential equation NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyUieUdGJy0lImZHNiMlInhH for all positive values of x, except possibly where NiMvJSJ4RyIiIg==, and is subject to the initial condition NiMvLSUieUc2IyIiIUYn. f := x -> piecewise(x<1,1,0): 'f(x)'=f(x); plot(f(x),x=0..3,thickness=2);

NiMvLSUiZkc2IyUieEctJSpQSUVDRVdJU0VHNiQ3JCIiIjJGJ0YsNyQiIiElKm90aGVyd2lzZUc=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

de := diff(y(x),x)+y(x)=f(x); ic := y(0)=0; desolveLN({de,ic},info=true); simplify(%); g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIkYqRi4tJSpQSUVDRVdJU0VHNiQ3JEYuMkYtRi43JCIiISUqb3RoZXJ3aXNlRw==NiM+JSNpY0cvLSUieUc2IyIiIUYpNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIkYpRi0tJSpQSUVDRVdJU0VHNiQ3JEYtMkYsRi03JCIiITFGLUYsNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiIlInhHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IyUieEc=NiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEdGJ0YpLSUkSW50RzYkLSUqUElFQ0VXSVNFRzYkNyRGKjJGKEYpNyQiIiExRilGKEYoNiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEdGJ0YpLCYtJSpQSUVDRVdJU0VHNiQ3JEYqMUYoRik3JC1GKzYjRikyRilGKEYpJiUiQ0dGNEYpNiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiEiIg==NiMlIUc=NiMvLSUieUc2IyUieEcsJi0lKlBJRUNFV0lTRUc2JDckIiIiMUYnRi03JC0lJGV4cEc2IywmRi1GLUYnISIiMkYtRidGLS1GMTYjLCRGJ0Y0RjQ=NiMvLSUieUc2IyUieEctJSpQSUVDRVdJU0VHNiQ3JCwmLSUkZXhwRzYjLCRGJyEiIkYxIiIiRjIxRidGMjckLCZGLUYxLUYuNiMsJkYyRjJGJ0YxRjIyRjJGJw==NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lKnBpZWNld2lzZUc2JjE5JCIiIiwmLSUkZXhwRzYjLCRGMCEiIkY3RjFGMTJGMUYwLCZGM0Y3LUY0NiMsJkYxRjFGMEY3RjFGKEYoRig=

Plotting the solution along with the gradient field suggests that the solution is correct.with(DEtools):with(plots): plot1:=DEplot(de,y(x),x=0..3,y=0..1,arrows=medium, color=blue): plot2:=plot(g(x),x=0..3,y=0..1,color=coral,thickness=2): display([plot1,plot2]);

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"3@5#>I*ps9&*F?7$$"3Unmmv2EHZF,$"3?J]ET0(>V*F?7$$"3!))>t#Q!>O6%F,$"3-SD\_$R>c*F?Fd^^l7'7$$"39$)*f0,X8*zF,Fb^^l7$$"3y1hyN-weBF?Fg^^l7$$"39>x)fWt=0#F?F\_^l7$$"3Gs$[AF4.*>F?Fa_^lFg_^l7'7$$"3'30x%p=3yBF?Fb^^l7$$"3/fr?/wqPRF?Fg^^l7$$"3pr(3W"3#3j$F?F\_^l7$$"3#[Up1kc#pNF?Fa_^lFd`^l7'7$$"37.")*yBHq&RF?Fb^^l7$$"397#GE(\l;bF?Fg^^l7$$"3zC)HG=o(4_F?F\_^l7$$"3#zZ!44S?[^F?Fa_^lFaa^l7'7$$"3Ac">jgwf`&F?Fb^^l7$$"37k#\5M-c4(F?Fg^^l7$$"3yw3D^br)y'F?F\_^l7$$"3#*H:^x8:FnF?Fa_^lF^b^l7'7$$"3K4-uuR#\6(F?Fb^^l7$$"3A<.Z4(\Xn)F?Fg^^l7$$"3))H>n>Hmn$)F?F\_^l7$$"3-$eKfu)41$)F?Fa_^lF[c^l7'7$$"3Si7;V8(Qp)F?Fb^^l7$$"39P"*y2(\`-"FhqFg^^l7$$"35%)H4)G5m%**F?F\_^l7$$"3APON9h/&))*F?Fa_^lFhc^l7'7$$"3qJq#f.g!z5Fhq$"3Od#yLJC>s*F?7$$"3>UC')ziYJ6Fhq$"3^mVn2TWD#*F?7$$"3*))H2v)>],6Fhq$"3X,_N!H*46$*F?7$$"3k!4NH+Q(Q6Fhq$"3Ax8Ig6xa$*F?Fgd^l7'7$$"3,P"pFxapB"FhqFed^l7$$"3]ZXq;5O*G"FhqFjd^l7$$"3?/%\Vs'Rf7FhqF_e^l7$$"3%f>x(RFj'H"FhqFde^lFje^l7'7$$"3KU7h4&\[R"FhqFed^l7$$"3#GlYNvbsW"FhqFjd^l7$$"3]4:>h9H<9FhqF_e^l7$$"3E,$>mZFXX"FhqFde^lFgf^l7'7$$"3kZLXYUu_:FhqFed^l7$$"37e()Q!\]^g"FhqFjd^l7$$"3#[hL!)>'=v:FhqF_e^l7$$"3d19Y8AU7;FhqFde^lFdg^l7'7$$"3%HX&H$)*Q1r"FhqFed^l7$$"3Wj3BF_/j<FhqFjd^l7$$"38?d([$43L<FhqF_e^l7$$"3)=^..&pJq<FhqFde^lFah^l7'7$$"3Cev8?P`o=FhqFed^l7$$"3uoH2k*R4#>FhqFjd^l7$$"3UDyrrc(4*=FhqF_e^l7$$"3><c9(o6#G>FhqFde^lF^i^l7'7$$"37j'zpXGk-#FhqFed^l7$$"31u]"4qM)y?FhqFjd^l7$$"3uI*f&3/()[?FhqF_e^l7$$"3GAx)RU1h3#FhqFde^lF[j^l7'7$$"3Uo<#Q>BV=#FhqFed^l7$$"3OzrvP%HnB#FhqFjd^l7$$"30O?SX^w1AFhqF_e^l7$$"3fF)H3;,SC#FhqFde^lFhj^l7'7$$"3ttQmIz@UBFhqFed^l7$$"3m%G*fuTi%R#FhqFjd^l7$$"3OTTC#))fYO#FhqF_e^l7$$"3!H$>n(*e*=S#FhqFde^lFe[_l7'7$$"30zf]nE6+DFhqFed^l7$$"3)**QT9"*=Db#FhqFjd^l7$$"3oYi3>YbADFhqF_e^l7$$"3AQS^M1zfDFhqFde^lFb\_l7'7$$"3O%3[VS2!eEFhqFed^l7$$"3H&\$G[OT5FFhqFjd^l7$$"3)>NGfN\/o#FhqF_e^l7$$"3_VhNr`o<FFhqFde^lF_]_l7'7$$"3n*=!>T@!f"GFhqFed^l7$$"3g+c7&Q3$oGFhqFjd^l7$$"3Hd/x#4W$QGFhqF_e^l7$$"3#)[#)>3,evGFhqFde^lF\^_l7'7$$"3)\HK!yoztHFhqFed^l7$$"3"fqn>7.i-$FhqFjd^l7$$"3giDhH)Qi*HFhqF_e^l7$$"39a./X[ZLIFhqFde^lFi^_l7'7$$!3si_5Uot%*yF,$"2c***************Fhq7$$"3si_5Uot%*yF,Ff__l7$$"3S=sP6b@wWF,$"3R&y:j_5U$**F?7$$"3'G@x8^:iZ%F,$"3d?%ot%*yl+"FhqFh__l7'Fh__l7$$"3#)y:j_5UoBF?Ff__l7$$"3Ru(e&>*ol-#F?F^`_l7$$"3$Qxe&>*ol-#F?Fc`_lFf`_l7'Ff`_l7$$"3kJE0@%ot%RF?Ff__l7$$"3[F)zzG;bg$F?F^`_l7$$"3PE)zzG;bg$F?Fc`_lF`a_l7'F`a_l7$$"3j$ot%*y:j_&F?Ff__l7$$"3Yz3ScOY%=&F?F^`_l7$$"3Oy3ScOY%=&F?Fc`_lFja_l7'Fja_l7$$"3sOZ*y:j_5(F?Ff__l7$$"3eK>#[-6Mw'F?F^`_l7$$"3YJ>#[-6Mw'F?Fc`_lFdb_l7'Fdb_l7$$"3#)*y:j_5Uo)F?Ff__l7$$"3n&)HC$ReBM)F?F^`_l7$$"3c%)HC$ReBM)F?Fc`_lF^c_l7'F^c_l7$FfqFf__l7$$"3xQSmhdI@**F?F^`_l7$$"3mPSmhdI@**F?Fc`_lFhc_l7'7$$"3CRg%z!yH!3"Fhq$"3E(p[*\`'\-"Fhq7$$"3lMM%y]G-8"Fhq$"3a=I^+lM](*F?7$$"3*3%pu"=%p+6Fhq$"3#39s&y^kP)*F?7$$"3c()*pm?U"Q6Fhq$"3m.L:&4a#z)*F?Fed_l7'7$$"3cW")yWD>Q7FhqFcd_l7$$"3'*RboWK7)G"FhqFhd_l7$$"3?Y!*e=*)ee7FhqF]e_l7$$"3'G47N%p.'H"FhqFbe_lFhe_l7'7$$"3')\-j"G(3'R"FhqFcd_l7$$"3EXw_")z,Y9FhqFhd_l7$$"3^^6VbO[;9FhqF]e_l7$$"3=)>a.oJRX"FhqFbe_lFef_l7'7$$"3=bBZ=?)Rb"FhqFcd_l7$$"3e](p$=F"Rg"FhqFhd_l7$$"3#oDtARyVd"FhqF]e_l7$$"3[.j><k#=h"FhqFbe_lFbg_l7'7$$"3\gWJbn(=r"FhqFcd_l7$$"3!f&=@bu!=w"FhqFhd_l7$$"38i`6HJFK<FhqF]e_l7$$"3!)3%QS:@(p<FhqFbe_lF_h_l7'7$$"3!ecc@\r(p=FhqFcd_l7$$"3?hR0#>-(>>FhqFhd_l7$$"3Wnu&f'y;!*=FhqF]e_l7$$"3590)3*ehF>FhqFbe_lF\i_l7'7$$"3mq')**GimF?FhqFcd_l7$$"3]mg*)Gpfx?FhqFhd_l7$$"3ws&*z-E1[?FhqF]e_l7$$"3U>EsF1^&3#FhqFbe_lFii_l7'7$$"3'fxSe'4c&=#FhqFcd_l7$$"3#=<Qdm"\NAFhqFhd_l7$$"31y;kRt&f?#FhqF]e_l7$$"3tCZck`SVAFhqFbe_lFfj_l7'7$$"3G")Go-dXVBFhqFcd_l7$$"37x-e-kQ$R#FhqFhd_l7$$"3O$y$[w?&QO#FhqF]e_l7$$"3.IoS,,I,CFhqFbe_lFc[`l7'7$$"3f')\_R/N,DFhqFcd_l7$$"3W#QA%R6G^DFhqFhd_l7$$"3o))eK8ou@DFhqF]e_l7$$"3MN*[#Q[>fDFhqFbe_lF`\`l7'7$$"3!>4nj<X#fEFhqFcd_l7$$"3v([ki(e<4FFhqFhd_l7$$"3)R*z;]:kzEFhqF]e_l7$$"3lS54v&*3<FFhqFbe_lF]]`l7'7$$"3@(>4K"*Rr"GFhqFcd_l7$$"31$f1Jhqq'GFhqFhd_l7$$"3I*45qGOv$GFhqF]e_l7$$"3'f9L>J%)\(GFhqFbe_lFj]`l7'7$$"3_-80]Y.vHFhqFcd_l7$$"3P)p[*\`'\-$FhqFhd_l7$$"3g/A&Q-Ja*HFhqF]e_l7$$"3F^_x[!zG.$FhqFbe_lFg^`l-%&COLORG6&%$RGBG$Fjq!""Fd_`l$"#5Fe_`l-F&6%7W7$FiqFiq7$$"3')*****\7t&pKF,$"3kt<7(Q+n@$F,7$$"3s******\i9RlF,$"37T%Q/;H*HjF,7$$"33++vVV)RQ*F,$"3=<tgKQ9d*)F,7$$"3/++vVA)GA"F?$"3a_5qgr1^6F?7$$"3+++]Peui=F?$"3WiTeLNa*p"F?7$$"3A++]i3&o]#F?$"3X7)e"*zDt@#F?7$$"3%)***\(oX*y9$F?$"3q(RW_du0q#F?7$$"3z***\P9CAu$F?$"3+om5r3w@JF?7$$"3!)***\P*zhdVF?$"3?WDs4BGKNF?7$$"31++v$>fS*\F?$"35"*y?^!*3JRF?7$$"3$)***\(=$f%GcF?$"3X(y7AZUTI%F?7$$"3Q+++Dy,"G'F?$"3gfO$yjiRm%F?7$$"33++]7<zboF?$"33P-k/k,i\F?7$$"3`+++v4&G](F?$"3G['pe)4ox_F?7$$"3!)*****\7nD:)F?$"3Cp'H5*GuubF?7$$"3[+++D!*oy()F?$"3N*y'QH?KVeF?7$$"3))***\PpnsM*F?$"3)p(\JL&oI2'F?7$$"3e+]P4_J&o*F?$"3T0*)eu()f.iF?7$$"3,++]siL-5Fhq$"3.S5=nZX1jF?7$$"3,++DJL(4."Fhq$"3([j)HXyTGhF?7$$"3-+++!R5'f5Fhq$"3CSXumqSbfF?7$$"3)***\P/QBE6Fhq$"3!4_Fu)ecrbF?7$$"3!******\"o?&="Fhq$"3cEr^OY\__F?7$$"31+]Pa&4*\7Fhq$"3'*\-VU^SB\F?7$$"33+]7j=_68Fhq$"3Z!*G!yf=#HYF?7$$"33++vVy!eP"Fhq$"3-US]s&))4M%F?7$$"34+](=WU[V"Fhq$"33j7RHx8#4%F?7$$"3)****\7B>&)\"Fhq$"32wz'eZ'oRQF?7$$"3)***\P>:mk:Fhq$"3K*[O#3w$Rf$F?7$$"3'***\iv&QAi"Fhq$"3)Hrhzr`GR$F?7$$"31++vtLU%o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Another way to solve this initial value problem is to make use of the Heaviside step function NiMvLSUiSEc2IyUieEctJSpQSUVDRVdJU0VHNiU3JCIiITJGJ0YsNyQlKnVuZGVmaW5lZEcvRidGLDckIiIiMkYsRic=.Maple has the Heaviside function built-in as Heaviside(x). 'H'(x)=convert(Heaviside(x),piecewise);

NiMvLSUiSEc2IyUieEctJSpQSUVDRVdJU0VHNiU3JCIiITJGJ0YsNyQlKnVuZGVmaW5lZEcvRidGLDckIiIiMkYsRic=

We can redefine NiMtJSJmRzYjJSJ4Rw== in terms of the Heaviside function (except where NiMvJSJ4RyIiIg==) as follows. f := x -> piecewise(x<1,1,0): 'f(x)'=f(x); alias(H=Heaviside): f := unapply(convert(f(x),Heaviside),x): 'f(x)'=f(x); plot(f(x),x=0..3,thickness=2);

NiMvLSUiZkc2IyUieEctJSpQSUVDRVdJU0VHNiQ3JCIiIjJGJ0YsNyQiIiElKm90aGVyd2lzZUc=NiMvLSUiZkc2IyUieEcsJiIiIkYpLSUiSEc2IywmRidGKUYpISIiRi4=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

Converting NiMtJSJmRzYjJSJ4Rw== back to piecewise form indicates that the new NiMtJSJmRzYjJSJ4Rw== is not defined at the discontinuity where NiMvJSJ4RyIiIg==.However, this does not affect the calculation significantly. convert(f(x),piecewise,x);

NiMtJSpQSUVDRVdJU0VHNiU3JCIiIjIlInhHRic3JCUqdW5kZWZpbmVkRy9GKUYnNyQiIiEyRidGKQ==

Here is the alternative solution. de := diff(y(x),x)+y(x) = f(x); ic := y(0)=0; desolveLN({de,ic},info=true); g := unapply(rhs(%),x);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIkYqRi4sJkYuRi4tJSJIRzYjLCZGLUYuRi4hIiJGNA==NiM+JSNpY0cvLSUieUc2IyIiIUYpNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIkYpRi0sJkYtRi0tJSJIRzYjLCZGLEYtRi0hIiJGMw==NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiIlInhHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IyUieEc=NiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEdGJ0YpLSUkSW50RzYkLCZGKkYpKiZGKkYpLSUiSEc2IywmRihGKUYpISIiRilGNUYoNiMlIUc=NiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEdGJ0YpLCpGKkYpKiZGKkYpLSUiSEc2IywmRihGKUYpISIiRilGMiomRi5GKS1GKzYjRilGKUYpJiUiQ0dGNUYpNiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiEiIg==NiMlIUc=NiMvLSUieUc2IyUieEcsKiIiIkYpLSUiSEc2IywmRidGKUYpISIiRi4qJi0lJGV4cEc2IywmRidGLkYpRilGKUYqRilGKS1GMTYjLCRGJ0YuRi4=NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwqIiIiRi0tJSJIRzYjLCY5JEYtRi0hIiJGMyomLSUkZXhwRzYjLCZGMkYzRi1GLUYtRi5GLUYtLUY2NiMsJEYyRjNGM0YoRihGKA==

Converting the new solution to piecewise form shows that this is essentially the same solution as before (except that the solution is not defined when x = 1).convert(g(x),piecewise);

NiMtJSpQSUVDRVdJU0VHNiU3JCwmLSUkZXhwRzYjLCQlInhHISIiRi0iIiJGLjJGLEYuNyQsJi1GKTYjRi1GLSUqdW5kZWZpbmVkR0YuL0YsRi43JCwmRihGLS1GKTYjLCZGLEYtRi5GLkYuMkYuRiw=

The graph of the new solution is the same as before.plot(g(x),x=0..3,y=0..1,color=coral,thickness=2);

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

;

NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomIiIjRiclInhHRidGJywqLSUkc2luRzYjJSJ0R0YnKiYtRi82IyomIiIkRidGMUYnRidGNkYpRicqJi1GLzYjKiYiIiZGJ0YxRidGJ0Y7RilGJyomLUYvNiMqJiIiKEYnRjFGJ0YnRkBGKUYn f := t -> sin(t)+sin(3*t)/3+sin(5*t)/5+sin(7*t)/7; de := diff(x(t),t)+2*x(t)=f(t); ic := x(0)=0; desolveLN({de,ic},info=true): g := unapply(rhs(%),t);

NiM+JSJmR2YqNiMlInRHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwqLSUkc2luRzYjOSQiIiIqJiNGMSIiJEYxLUYuNiMsJEYwRjRGMUYxKiYjRjEiIiZGMS1GLjYjLCRGMEY6RjFGMSomI0YxIiIoRjEtRi42IywkRjBGQEYxRjFGKEYoRig=NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieEc2IyUidEdGLSIiIiomIiIjRi5GKkYuRi4sKi0lJHNpbkdGLEYuKiYjRi4iIiRGLi1GMzYjLCRGLUY2Ri5GLiomI0YuIiImRi4tRjM2IywkRi1GPEYuRi4qJiNGLiIiKEYuLUYzNiMsJEYtRkJGLkYuNiM+JSNpY0cvLSUieEc2IyIiIUYpNiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieEc2IyUidEdGLCIiIiomIiIjRi1GKUYtRi0sKi0lJHNpbkdGK0YtKiYjRi0iIiRGLS1GMjYjLCRGLEY1Ri1GLSomI0YtIiImRi0tRjI2IywkRixGO0YtRi0qJiNGLSIiKEYtLUYyNiMsJEYsRkFGLUYtNiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQiIiMlInRHNiQlNX5+fn5+fn5+fn5+fn5+fn49fn5+Ry0lJGV4cEc2IywkJSJ0RyIiIw==NiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKS0lJEludEc2JCwkKiYsKi0lJHNpbkdGJyIkMCIqJiIjTkYpLUY2NiMsJEYoIiIkRilGKSomIiNARiktRjY2IywkRigiIiZGKUYpKiYiIzpGKS1GNjYjLCRGKCIiKEYpRilGKUYqRikjRilGN0YoNiMlIUc=NiMvKiYtJSJ4RzYjJSJ0RyIiIi0lJGV4cEc2IywkRigiIiNGKSw0KiYtJSRjb3NHRidGKUYqRikjISIiIiImKigjRi5GNUYpRipGKS0lJHNpbkdGJ0YpRikqJiNGKSIjOEYpKiYtRjI2IywkRigiIiRGKUYqRilGKUY0KigjRi4iI1JGKUYqRiktRjlGP0YpRikqJiNGKSIjSEYpKiZGKkYpLUYyNiMsJEYoRjVGKUYpRjQqKCNGLiIkWCJGKUYqRiktRjlGS0YpRikqJiNGKSIjYEYpKiZGKkYpLUYyNiMsJEYoIiIoRilGKUY0KigjRi4iJHIkRilGKkYpLUY5RlZGKUYpJiUiQ0c2I0YpRik=NiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiMiJicqSCQiJjAqKio=NiMlIUc=NiM+JSJnR2YqNiMlInRHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCw0LSUkY29zRzYjOSQjISIiIiImKiYjIiIjRjMiIiItJSRzaW5HRi9GN0Y3KiYjRjciIzhGNy1GLjYjLCRGMCIiJEY3RjIqJiNGNiIjUkY3LUY5Rj5GN0Y3KiYjRjciI0hGNy1GLjYjLCRGMEYzRjdGMiomI0Y2IiRYIkY3LUY5RklGN0Y3KiYjRjciI2BGNy1GLjYjLCRGMCIiKEY3RjIqJiNGNiIkciRGNy1GOUZTRjdGNyomIyImJypIJCImMCoqKkY3LSUkZXhwRzYjLCRGMCEiI0Y3RjdGKEYoRig=

plot([f(t),g(t)],t=0..10);

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;

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUiZkc2IyUieEdGJyUieUdGJ0YnLSUiZ0dGLS8tRi82IyUjeDBHJSN5MEc= f := 'f': g := 'g': de := diff(y(x),x)+f(x)*y(x)=g(x); ic := y(x0)=y0; sol := desolveLN({de,ic},info=true);

NiM+JSNkZUcvLCYtJSVkaWZmRzYkLSUieUc2IyUieEdGLSIiIiomLSUiZkdGLEYuRipGLkYuLSUiZ0dGLA==NiM+JSNpY0cvLSUieUc2IyUjeDBHJSN5MEc=NiMlIUc=NiQlMExpbmVhcn5ERX4ufi5+fkcvLCYtJSVEaWZmRzYkLSUieUc2IyUieEdGLCIiIiomLSUiZkdGK0YtRilGLUYtLSUiZ0dGKw==NiMlIUc=NiQlOUludGVncmF0aW5nfmZhY3Rvcn4ufi5+fkctJSRleHBHNiMtJSRJbnRHNiQtJSJmRzYjJSJ4R0YtNiMvKiYtJSJ5RzYjJSJ4RyIiIi0lJGV4cEc2Iy0lJGludEc2JC0lImZHNiMlI191Ry9GMzslIUdGKEYpLCYtRi42JComLSUiZ0c2IyUjX3ZHRiktRis2Iy1GLjYkRjAvRjM7RjZGPkYpL0Y+RjVGKSYlIkNHNiNGKUYpNiMlIUc=NiQlRUFwcGx5aW5nfnRoZX5pbml0aWFsfmNvbmRpdGlvbn4ufi5+fkcvJiUiQ0c2IyIiIiwmKiYlI3kwR0YoLSUkZXhwRzYjLSUkSW50RzYkLSUiZkc2IyUieEcvRjU7JSFHJSN4MEdGKEYoLUYwNiQqJi0lImdHNiMlI192R0YoLUYtNiMtJSRpbnRHNiQtRjM2IyUjX3VHL0ZIO0Y4RkBGKC9GQEY3ISIiNiMlIUc=NiM+JSRzb2xHLy0lInlHNiMlInhHKiYsJi0lJEludEc2JComLSUiZ0c2IyUjX3ZHIiIiLSUkZXhwRzYjLSUkaW50RzYkLSUiZkc2IyUjX3VHL0Y+OyUhR0YzRjQvRjM7JSN4MEdGKUY0KiYlI3kwR0Y0LUY2NiMtRi02JC1GPEYoL0YpO0ZBRkRGNEY0RjQtRjY2Iy1GOTYkRjsvRj47RkFGKSEiIg==

simplify(subs(sol,de));

NiMvLSUiZ0c2IyUieEdGJA==

However, Maple's solution does not check in the differential equation.sol2 := dsolve({de,ic},y(x)); simplify(subs(sol2,de));

NiM+JSVzb2wyRy8tJSJ5RzYjJSJ4RyomLCYtJSRJbnRHNiQqJi0lImdHNiMlInVHIiIiLSUkZXhwRzYjLUYtNiQtJSJmR0YyRjNGNC9GMzslI3gwR0YpRjQlI3kwR0Y0RjQtRjY2Iy1GLTYkLCRGOiEiIkY8RjQ=NiMvKiYtJSJnRzYjJSJ4RyIiIi0lJGV4cEc2IywmLSUkSW50RzYkLSUiZkdGJ0YoRiktRi82JC1GMjYjJSJ1Ry9GNzslI3gwR0YoISIiRilGJQ==

;

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In questions 1 to 8, solve the given initial value problem.In each case plot the graph of the solution along with the associated direction field for the differential equation.Note: When setting up the differential equation to be solved by either dsolve or desolveLN: Use diff(y(x),x) to replace the mathematical symbol NiMqJiUjZHlHIiIiJSNkeEchIiI=.Replace y by y(x) everywhere y appears in the differential equation.;

NiQvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyUieUdGJy0lJHNpbkc2IyUieEcvLUYqNiMiIiFGMg== __________________________________________________________________________;

NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooJSJ4R0YnJSJ5R0YnLCYqJEYrIiIjRiciIipGJ0YpRiciIiE=, NiMvLSUieUc2IyIiISIiIg== __________________________________________________________________________;

NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRiclInhHRiclInlHRidGJ0Ys, NiMvLSUieUc2IyIiISIiIw== __________________________________________________________________________;

x NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyUieUdGJyomJSJ4R0YnLSUkc2luRzYjRixGJw==, NiMvLSUieUc2IyomJSNQaUciIiIiIiMhIiJGKQ== __________________________________________________________________________;

NiMtJSFHNiMsJiIiIkYnJSJ4R0Yn NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomJSJ4R0YnJSJ5R0YnRikqJEYrIiIj, NiMvLSUieUc2IyIiISIiIg== __________________________________________________________________________;

NiMtJSFHNiMsJiIiIkYnKiQlInhHIiIjRic= NiMvLCoqJiUjZHlHIiIiJSNkeEchIiJGJyomJSJ4R0YnJSJ5R0YnRidGK0YnKiRGKyIiJEYnIiIh, NiMvLSUieUc2IyIiISIiIw== __________________________________________________________________________;

x NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLCZGJ0YnJSJ4R0YnRiclInlHRidGJyomLSUkZXhwRzYjLCRGLEYpRictJSRzaW5HNiMqJiIiI0YnRixGJ0Yn, y(1) = 0 __________________________________________________________________________;

NiMtJSFHNiMsJiIiIkYnLSUkZXhwRzYjJSJ4R0Yn NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyomLSUkZXhwRzYjJSJ4R0YnJSJ5R0YnRicqJEYuIiIl, y(0) = 2 __________________________________________________________________________;

Given NiMvLSUiZkc2IyUieEctJSpQSUVDRVdJU0VHNiQ3JEYnMUYnIiIjNyQiIiEyRi1GJw==, find a continuous solution which satisfies the differential equation NiMvLCYqJiUjZHlHIiIiJSNkeEchIiJGJyooIiIjRiclInhHRiclInlHRidGJy0lImZHNiNGLA== for all positive values of x, except possibly where NiMvJSJ4RyIiIw==, and is subject to the initial condition NiMvLSUieUc2IyIiISwkIiIiISIi. __________________________________________________________________________;

(a) Construct a first order linear differential equation, for which dsolve or desolveLN can find the general solution.(b) For the differential equation chosen in (a) pick an initial condition and find the particular solution with this initial condition. (c) Plot the graph of the particular solution along with the direction field associated with the differential equation.__________________________________________________________________________;

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