> #sfb Numerical DE solver for ode's using maple
> # we want to solve <x', y'> = <-y, x> numberically
> # It is easy to check it has a solution < a cos(t), a sin(t) >
> with(plots):with(DEtools):
> ODE1:=diff(x(t),t)=-y(t);
> ODE2:=diff(y(t),t)=x(t);
> SYS:=[ODE1,ODE2];
> VARS:=[x(t),y(t)];
> I_C:=[[x(0)=1,y(0)=0]];
> T_RANGE:=t=0..6;
> STEP:=stepsize=0.1;
> XvsT:=DEplot(SYS,VARS,T_RANGE,I_C,STEP,scene=[t,x(t)],linecolor=blue):
> YvsT:=DEplot(SYS,VARS,T_RANGE,I_C,STEP,scene=[t,y(t)],linecolor=red):
> XvsY:=DEplot(SYS,VARS,T_RANGE,I_C,STEP,scene=[x(t),y(t)],linecolor=green,color=red):
> #The next comand "displays" the first two plots together
> display(XvsT,YvsT,title="Solution vs time");
> #The next plot is constrained so the x y scales are the same
> display(XvsY,title="Trajectory --- Phase space",scaling=constrained);
> # compare the "good" solver with "Euler -- dumb version" Forward Euler
> DumbXvsT:=DEplot(SYS,VARS,T_RANGE,I_C,scene=[t,x(t)],linecolor=yellow,method=classical[foreuler]):
> display(XvsT,DumbXvsT,title="Yellow Line is Euler (unimproved)");
> DumbXvsY:=DEplot(SYS,VARS,T_RANGE,I_C,scene=[x(t),y(t)],linecolor=yellow,method=classical[foreuler]):
> display(XvsY,DumbXvsY,title="Phase plane: Yellow Line is Euler (unimproved)",scaling=constrained);
> # add a text label to the YvsT at first max (Pi/2,1) and moved it slightly
> display(YvsT,textplot([Pi/2+0.1,1,"First Max at (Pi/2,1)"],align={ABOVE,RIGHT}));
>