Differential equations involving the function composition

[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?

 

[mbp]

It looks like you can apply separation of variables and integrate

 

[pasmith]

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)).

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

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

Such ODEs are in general non-linear, and there are no general strategies for solving non-linear ODEs unless they happen to be separable.

 

[dftfunctional]

It looks like you can apply separation of variables and integrate

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

Thank you both,


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

It looks like you can apply separation of variables and integrate

Such ODEs are in general non-linear, and there are no general strategies for solving non-linear ODEs unless they happen to be separable.

If it is separable, how I can proceed to finding a solution considering that there is the composition given in the function?

 

[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

 

[dftfunctional]

Multiply both sides for dx getting


Then integrate both sides

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

where G is a primitive of g

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