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A System of Three ODEs

  1. May 7, 2010 #1
    1. The problem statement, all variables and given/known data
    In a problem I was given a system of three differential equations concerning three functions, x(t), y(t) and z(t):

    dx(t)=2y(t)dt,
    dy(t)=[z(t)-x(t)]dt,
    dz(t)=[c^2x(t)-2y(t)]dt. (where c is a constant)

    The problem asked me to prove that when t is large, x(t)+z(t) converges to K*exp{wt},
    where w is a positive real root of equation w^3+4w-2c^2=0,

    2. Relevant equations



    3. The attempt at a solution

    I haven't studied how to solve this kind of ODE system in my calcus class, so now I am stuck at the beginning of this question. If you are willing to take time to look at it for me, I will be real grateful for that. Thanks!
     
  2. jcsd
  3. May 7, 2010 #2

    vela

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    The middle equation is

    y'(t) = z(t)-x(t).

    If you differentiate it, you'll get

    y''(t) = z'(t)-x'(t)

    You can substitute for x'(t) and z'(t) using the first and third equations. With a bit more manipulation, you can eventually get a differential equation for just y(t), which you should be able to solve (in principle). Next, write (x+z)'' in terms of y. Then you should be able to argue what you're trying to show.
     
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