MaplePrimes - answers and comments on Question, Solving a non-linear differential equation

series solution

jakubi — Thu, 02 Jul 2009 00:42:54 Z

You may try a series solution like:

de:= diff(y(t),t)=cos(Pi*t*y(t)):
ics:=y(0)=0:
dsolve({de,ics},y(t),series);
                                   2
                                 Pi   5      6
                      y(t) = t - --- t  + O(t )
                                 10

Note: use Pi for 3.14...

de

Robert Israel — Thu, 02 Jul 2009 01:40:14 Z

In all likelihood there are no closed-form solutions. Besides series, you could try numerical solutions. Using Pi, as jakubi remarked:

> S:= dsolve({de, ics}, y(t), numeric);

> plots[odeplot](S,[t,y(t)],t=0..10);

It looks like y(t) ~ 1/(2 t) as t -> infinity. More precisely:

1/(2*t)+1/(2*Pi*t^3)+3/2/Pi^2/t^5+1/48*(Pi^2+360)/Pi^3/t^7+1/6*(2*Pi^2+315)/Pi^4/t^9+3/1280*(Pi^4+1920*Pi^2+201600)/Pi^5/t^11 + O(t^(-13))

Oo thank you! That

Melawrgh — Thu, 02 Jul 2009 03:59:16 Z

Oo thank you! That helped.

Now I'm trying to graphs a sequence of solutions with different initial conditions on one graph. So I wrote this:

display(seq(plots[odeplot](dsolve({de, y(0)=k}, y(t), numeric), [t,y(t)], t=0..10), k=0..4, 0.2), insequence=true)

But the output from that is basically:

display(PLOT(...), PLOT(...).... etcetc)

I checked the brackets and I think they're all okay. Is it the syntax? Or just the idea in general?

display

Robert Israel — Thu, 02 Jul 2009 04:35:45 Z

The display command is in the plots package. So you could use plots[display] instead of just display.

The option insequence=true will give you an animation. Leave it out if you want the different solutions all on one graph.