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        riddles >> hard >> differential equation
        
(Message started by: hparty on Sep 21^st, 2010, 5:06pm)

Title: differential equation
Post by hparty on Sep 21^st, 2010, 5:06pm

Consider the second order differential equations

(1) y''(x) + y(x) +y^3(x) = 0 , x \in R

(2) y''(x) + y'(x)+y(x) +y^3(x) = 0, x\in R

Prove that (1) has a solution for all x \in R.
What about the equation in (2).

Thanks for help in advance

Title: Re: differential equation
Post by hparty on Sep 23^rd, 2010, 5:34pm

Any hint?? :-/

Title: Re: differential equation
Post by Aryabhatta on Sep 23^rd, 2010, 11:11pm

y(x) = 0 is a trivial solution to both.

To get a non-trivial solution to 1) the below might help.

Multiply by 2y'(x).

You get

d( (y')^2)/dx = -2y'(y + y^3)

Now integrate, you get

(y')^2 = -y^2 - y^4/2 + C

Thus

y' = sqrt(C - y^2 -y^4/2)

i.e.

y'/(sqrt(C-y^2 -y^4/2) = 1

Integrate again (I have no clue how to do that).

Perhaps it will go somewhere.

Didn't think about 2).

Title: Re: differential equation
Post by towr on Sep 24^th, 2010, 4:03am

Perhaps there is a non-constructive proof? I.e. without having to find the solution. (Although wolframalpha does give one, but it involves elliptic curves.)
For example, you can interpret the first equation as describing a physical system F(t) = - [s(t) + s(t)^3]. I don't think there's a reason why such a system couldn't exist, with an opposing force growing as a third degree polynomial of the distance from equilibrium position.

Title: Re: differential equation
Post by Aryabhatta on Sep 24^th, 2010, 10:27am

on 09/24/10 at 04:03:19, towr wrote:

Yup, I am pretty sure there will be some existence theorem which will imply the existence of a non-trivial solution. No idea what though :P