Runge-Kutta and Euler's Method Using Maple

Some Comparisons

>	with(plots):

Warning, the name changecoords has been redefined

>	eq1:=diff(y(x),x)=cos(x);

>	ini1:=y(0)=0;

Here is the analytical solution.

>	dsolve({eq1,ini1},{y(x)});

>	AnalSoln:=plot(sin(x),x=0..2*Pi,y=-1.5..1.5,color=blue,thickness=2):

Here is a Runge-Kutta solution (by default).

>	sol:=dsolve({eq1,ini1},{y(x)},type=numeric);

>	RungeKutta:=odeplot(sol,[x,y(x)],0..2*Pi,thickness=2):

>	sol(Pi/2);

It is difficult to visually distinguish between the Runge-Kutta solution and the analytical solution.

>	display(AnalSoln,RungeKutta);

Here are some Euler solutions.

>	sol2:=dsolve({eq1,ini1},{y(x)},type=numeric,method=classical[foreuler],stepsize=Pi/32);

>	EulerSoln2:=odeplot(sol2,[x,y(x)],0..2*Pi,thickness=2):

>	sol2(Pi/2);

>	display(AnalSoln,EulerSoln2);

>	sol3:=dsolve({eq1,ini1},{y(x)},type=numeric,method=classical[foreuler],stepsize=Pi/64);

>	EulerSoln3:=odeplot(sol3,[x,y(x)],0..2*Pi,thickness=2):

>	sol3(Pi/2);

>	display(AnalSoln,EulerSoln3);

>	sol4:=dsolve({eq1,ini1},{y(x)},type=numeric,method=classical[foreuler],stepsize=Pi/128);

>	EulerSoln4:=odeplot(sol4,[x,y(x)],0..2*Pi,thickness=2):

>	sol4(Pi/2);

>	display(AnalSoln,EulerSoln4);

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