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Differential equations involving the function composition

  1. Nov 28, 2013 #1
    I have not met differential equations involving the composition functions (also not much literature on it).

    Assume we know the form of g=g(x), and need to solve the following differential equation, finding f=f(x):


    Where g∘f=g(f(x)).

    Does anybody have a strategy for solving the above-mentioned differential equation, that involves such function composition?
  2. jcsd
  3. Nov 28, 2013 #2


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    It looks like you can apply separation of variables and integrate
  4. Nov 28, 2013 #3


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    Homework Helper

    By inspection, [itex]f(x) = x[/itex] is a solution.

    Such ODEs are in general non-linear, and there are no general strategies for solving non-linear ODEs unless they happen to be separable.
  5. Nov 28, 2013 #4

    Thank you both,

    As a trained material scientist I am not an expert on ODE. Could you please, therefore provide me more details.

    If it is separable, how I can proceed to finding a solution considering that there is the composition given in the function?
  6. Nov 28, 2013 #5


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    Multiply both sides for dx getting

    g(f(x)) f'(x) dx = g(x) dx

    Then integrate both sides

    G(f(x)) = G(x)

    where G is a primitive of g, and you get f(x) = x as pasmith suggested
  7. Nov 28, 2013 #6

    Thank you very much,

    as far as I understood G(f(x)) = G(x) would be equivalent to:

    ∫ g(f(x)df = ∫g(x)dx

    And per inspection we could find that one solution is f(x)=x. Empirically I know that there are many solutions to the given equation. Is there a way for "exctracting" f(x) or getting rid of the composition function in the integrand from the (again):

    ∫ g(f(x)df = ∫g(x)dx
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