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riddles >> putnam exam (pure math) >> Zero?
(Message started by: kyle1080 on Nov 9th, 2009, 2:56pm)

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Title: Zero?
Post by kyle1080 on Nov 9th, 2009, 2:56pm
Prove or disprove that all solutions of x"+|x'|x'+x3=0 go to zero as t->\infinity.

Title: Re: Zero?
Post by Michael Dagg on Nov 13th, 2009, 4:51pm
Hint: [hide]Rewrite the ODE in terms of phase plane variables so that
it can be integrated with respect to  x . Pick a line segment
with endpoints lying along some trajectory and then argue that
the path of the function g(x,y) = C (constant) obtained by
integration closes in on the origin whereby the trajectory
crosses g(x,y) successively.

Since the trajectory is arbitrary you're done.[/hide]

Title: Re: Zero?
Post by kyle1080 on Nov 14th, 2009, 3:36pm
Don't follow. Differentiation is with respect to t not x. Problem is not that simple.

Title: Re: Zero?
Post by Michael Dagg on Nov 15th, 2009, 6:59am
[hide] Note that

x" = d^2/dt^2[x] = 1/2*d/dx[(x')^2] .

Then

1/2*d/dx[(x')^2] + x^3 = -|x'|x'

but in the phase plane  dx = x' dt = y dt , that is, y = dx/dt = x'
and so

1/2*d/dx[y^2] + x^3 = -|y|y .
[/hide]

Title: Re: Zero?
Post by kyle1080 on Nov 15th, 2009, 10:45am
Relation for x'' is a surprise. I still don't get it. There is no function in the problem having the independent variable x, unless you use x3 as function, like f(x)=x3.

Title: Re: Zero?
Post by diemert on Nov 21st, 2009, 11:01am
OR x=0

Title: Re: Zero?
Post by Michael Dagg on Nov 23rd, 2009, 8:52am
> OR x=0

True, but the trivial solution is not representive of
all solutions.

> There is no function in the problem having the
> independent variable x, unless you use x3 as
> function, like f(x)=x3.

Not necessary. You may want to review differentiation
and integration.

Title: Re: Zero?
Post by kyle1080 on Nov 25th, 2009, 10:32am
That is a setback. Do you know what your are talking about? The right side of the equation doesn't integrate with respect to x as far I can see and I don't see how to relate an expression that does with one that doesn't.

Title: Re: Zero?
Post by SMQ on Nov 25th, 2009, 10:38am

on 11/25/09 at 10:32:56, kyle1080 wrote:

While I haven't followed the details of this thread, it has been my experience that yes, Michael Dagg knows what he's talking about--and better than most.

--SMQ

Title: Re: Zero?
Post by ThudanBlunder on Nov 25th, 2009, 1:27pm
Probably Kyle has not studied phase planes yet.

Title: Re: Zero?
Post by kyle1080 on Nov 25th, 2009, 5:28pm
Hello! I am friendly and sincere. I wasn't implying anything specific. Please don't take what I said out of context. Yes, I know about the phase plane but I don't understand the integration and what was said for the conclusion. I am listening.

Title: Re: Zero?
Post by Aryabhatta on Dec 21st, 2009, 11:20pm

on 11/25/09 at 17:28:54, kyle1080 wrote:
 Hello! I am friendly and sincere. I wasn't implying anything specific. Please don't take what I said out of context. Yes, I know about the phase plane but I don't understand the integration and what was said for the conclusion. I am listening.

The first sentence of Michael Dagg's hint says:

Rewrite the ODE in terms of phase plane variables so that it can be integrated with respect to  x.

Did you manage to get past this or are you stuck at this point?