Differential equations involving the function composition

1. Nov 28, 2013

dftfunctional

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):

(g∘f)f'=g

Where g∘f=g(f(x)).

Does anybody have a strategy for solving the above-mentioned differential equation, that involves such function composition?

2. Nov 28, 2013

mbp

It looks like you can apply separation of variables and integrate

3. Nov 28, 2013

pasmith

By inspection, $f(x) = x$ 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.

4. Nov 28, 2013

dftfunctional

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?

5. Nov 28, 2013

mbp

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

6. Nov 28, 2013

dftfunctional

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