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Nonautonomous ode

  1. Oct 15, 2006 #1
    I need some help with a differential equation:

    dy/dt = A*y^n + g(t)

    where A and n are constants that can be any real numbers and g(t) is just
    some function of t (e.g., g(t) = exp(-t/tau) or g(t) = constant or g(t) =
    0). I think this is a nonlinear ODE.
    If g(t) is a constant then it is an "autonomous" equation but if g(t) depends explicitly on t then it is "nonautonomous".
    I can solve it if g(t) = 0. and g(t) = constant.
    I can't find a solution if g(t) is some arbitrary function of t.

    Any help would be much appreciated!
  2. jcsd
  3. Oct 16, 2006 #2


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    Yes, because of the "y^n" that's a non-linear equation. There is no general way of solving such equations. The solution will depend strongly on exactly what g(t) is.
  4. Oct 16, 2006 #3


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    Note for example if you choose A=1, n=2, g(t)=1, then your solution will blow up at a FINITE time. Other choices won't have this "pathological" behaviour, so the precise shape of g is crucial for saying anything particular about the solution of your diff.eq.
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