second order ODE
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From: Joubert on 9 Jun 2010 14:35 I'm supposed to study existence of periodic and global C^infty solutions of the following ODE: y'' = a^2 - y^2 where a is a real parameter. How would you go about this one?
From: Zdislav V. Kovarik on 9 Jun 2010 15:49 On Wed, 9 Jun 2010, Joubert wrote: > I'm supposed to study existence of periodic and global C^infty solutions > of the following ODE: > > y'' = a^2 - y^2 > > where a is a real parameter. > > How would you go about this one? > The two obvious constant solutions are y=a and y=-a. For non-constant solutions, multiply both sides by (2*y') and you get a first integral (y')^2 = 2*E + 2*a^2*y - (2/3)*y^3 Then it becomes messy, discussing the non-negativity of the right-hand side. How much of elliptic function theory and results can you use? Cheers, ZVK(Slavek).
From: Joubert on 9 Jun 2010 17:49 On 06/09/2010 09:49 PM, Zdislav V. Kovarik wrote: > Then it becomes messy, discussing the non-negativity of the right-hand > side. How much of elliptic function theory and results can you use? > Hi, pretty much zero elliptic function theory. You said I should discuss the sign of the righthand side right? Once I rule out the intervals in which it is negative what do I do? Thanks for the answer.
From: Robert Israel on 9 Jun 2010 21:24
On Wed, 09 Jun 2010 20:35:18 +0200, Joubert wrote: > I'm supposed to study existence of periodic and global C^infty solutions > of the following ODE: > > y'' = a^2 - y^2 > > where a is a real parameter. > > How would you go about this one? Phase plane analysis for the system y' = v, v' = a^2 - y^2. You might also note that you can assume wlog that a = 0 or 1 (consider scaling y and t). -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |