wu :: forums
« wu :: forums - Zero? »

Welcome, Guest. Please Login or Register.
Dec 5th, 2024, 3:22am

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   riddles
   putnam exam (pure math)
(Moderators: Grimbal, Eigenray, towr, william wu, Icarus, SMQ)
   Zero?
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: Zero?  (Read 8711 times)
kyle1080
Newbie
*





   


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

Prove or disprove that all solutions of x"+|x'|x'+x3=0 go to zero as t->\infinity.
IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: Zero?  
« Reply #1 on: Nov 13th, 2009, 4:51pm »
Quote Quote Modify 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.
IP Logged

Regards,
Michael Dagg
kyle1080
Newbie
*





   


Posts: 6
Re: Zero?  
« Reply #2 on: Nov 14th, 2009, 3:36pm »
Quote Quote Modify Modify

Don't follow. Differentiation is with respect to t not x. Problem is not that simple.
IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: Zero?  
« Reply #3 on: Nov 15th, 2009, 6:59am »
Quote Quote Modify 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 .
IP Logged

Regards,
Michael Dagg
kyle1080
Newbie
*





   


Posts: 6
Re: Zero?  
« Reply #4 on: Nov 15th, 2009, 10:45am »
Quote Quote Modify 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.
IP Logged
diemert
Newbie
*





   


Gender: male
Posts: 29
Re: Zero?  
« Reply #5 on: Nov 21st, 2009, 11:01am »
Quote Quote Modify Modify

OR x=0
IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: Zero?  
« Reply #6 on: Nov 23rd, 2009, 8:52am »
Quote Quote Modify 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.
IP Logged

Regards,
Michael Dagg
kyle1080
Newbie
*





   


Posts: 6
Re: Zero?  
« Reply #7 on: Nov 25th, 2009, 10:32am »
Quote Quote Modify 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.
IP Logged
SMQ
wu::riddles Moderator
Uberpuzzler
*****






   


Gender: male
Posts: 2084
Re: Zero?  
« Reply #8 on: Nov 25th, 2009, 10:38am »
Quote Quote Modify 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
IP Logged

--SMQ

ThudnBlunder
Uberpuzzler
*****




The dewdrop slides into the shining Sea

   


Gender: male
Posts: 4489
Re: Zero?  
« Reply #9 on: Nov 25th, 2009, 1:27pm »
Quote Quote Modify Modify

Probably Kyle has not studied phase planes yet.
IP Logged

THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
kyle1080
Newbie
*





   


Posts: 6
Re: Zero?  
« Reply #10 on: Nov 25th, 2009, 5:28pm »
Quote Quote Modify 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.
IP Logged
Aryabhatta
Uberpuzzler
*****






   


Gender: male
Posts: 1321
Re: Zero?  
« Reply #11 on: Dec 21st, 2009, 11:20pm »
Quote Quote Modify 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?
 
 
IP Logged
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board