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kyle1080
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Posts: 6
 Zero?   « on: Nov 9th, 2009, 2:56pm » Quote Modify

Prove or disprove that all solutions of x"+|x'|x'+x3=0 go to zero as t->\infinity.
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Michael Dagg
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 Re: Zero?   « Reply #1 on: Nov 13th, 2009, 4:51pm » Quote Modify

Hint: 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.
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Regards,
Michael Dagg
kyle1080
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Posts: 6
 Re: Zero?   « Reply #2 on: Nov 14th, 2009, 3:36pm » Quote Modify

Don't follow. Differentiation is with respect to t not x. Problem is not that simple.
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Michael Dagg
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 Re: Zero?   « Reply #3 on: Nov 15th, 2009, 6:59am » Quote Modify

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 .
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Regards,
Michael Dagg
kyle1080
Newbie

Posts: 6
 Re: Zero?   « Reply #4 on: Nov 15th, 2009, 10:45am » Quote Modify

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.
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diemert
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 Re: Zero?   « Reply #5 on: Nov 21st, 2009, 11:01am » Quote Modify

OR x=0
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Michael Dagg
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 Re: Zero?   « Reply #6 on: Nov 23rd, 2009, 8:52am » Quote Modify

> 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.
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Regards,
Michael Dagg
kyle1080
Newbie

Posts: 6
 Re: Zero?   « Reply #7 on: Nov 25th, 2009, 10:32am » Quote Modify

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.
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SMQ
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 Re: Zero?   « Reply #8 on: Nov 25th, 2009, 10:38am » Quote Modify

on Nov 25th, 2009, 10:32am, kyle1080 wrote:
 [...] Do you know what your are talking about? [...]

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
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--SMQ

ThudnBlunder
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 Re: Zero?   « Reply #9 on: Nov 25th, 2009, 1:27pm » Quote Modify

Probably Kyle has not studied phase planes yet.
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kyle1080
Newbie

Posts: 6
 Re: Zero?   « Reply #10 on: Nov 25th, 2009, 5:28pm » Quote Modify

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.
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Aryabhatta
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 Re: Zero?   « Reply #11 on: Dec 21st, 2009, 11:20pm » Quote Modify

on Nov 25th, 2009, 5:28pm, 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?

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