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Solving a non-linear differential equation

Posted on 2009-07-01 14:25 By Melawrgh (2)
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Hi!

I'm trying to solve a differential equation of the form y'=cos(pi*t*y).

I wrote:

<de:= diff(y(t),t)=cos(pi*t*y(t))

And that works fine.

Then I tried to use dsolve(de) but that shows nothing and just goes to the next line

So I figured maybe it couldn't solve it and decided to try the transform methods. For instance,

with(inttrans)

ics:=y(0)=0

dsolve({de,ics},y(t),method=laplace) <- with regards to this, I also tried Fourier

But it gives me an error message: Error, (in dsolve/inttrans/solveit) transform method can only be applied to linear ODE

 

I don't know what to do with it... Thanks in advance for any help :)

‹ solve please..... roots of trig func ›

series solution

Comment on 2009-07-01 16:42 from jakubi (Maple Rating 4 1567)

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...

Robert Israel's picture

de

Comment on 2009-07-01 17:40 from Robert Israel (Maple Rating 4 2434)

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

Comment on 2009-07-01 19:59 from Melawrgh (2)

<p>Oo thank you! That helped.</p>
<p>Now I'm trying to graphs a sequence of solutions with different initial conditions on one graph. So I wrote this:</p>
<p>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)</p>
<p>&nbsp;</p>
<p>But the output from that is basically:</p>
<p>display(PLOT(...), PLOT(...).... etcetc)</p>
<p>&nbsp;</p>
<p>I checked the brackets and I think they're all okay. Is it the syntax? Or just the idea in general?</p>

Robert Israel's picture

display

Comment on 2009-07-01 20:35 from Robert Israel (Maple Rating 4 2434)

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.

 

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