Can Lagrange Differential Equations Involve Fourier Transforms?

[jbowers9]

Homework Statement

A Lagrange differential eq. represented as follows:

y = xf(y') + g(y')

Let y' = P
and after some fancy footwork;

dx/dP - xf'(P)/(P - f(P)) = g'(P)/(P - f(P)

Homework Equations

Now, the link that I got this from states that this is a 1st ode in standard form.
When I attempt to solve such I end up w/the following integrand:

$$\int f'(P)dP/(P-f(P))$$


I have 2 questions. How do I proceed, and when I do, is this going to involve the natural log of a Fourier transform?

 

[lippyka]

The Attempt at a SolutionI know that this is a separable ode, so I can use the following formula: \int f'(P)dP/(P-f(P)) = \int f'(P)dP - \int f'(P)f(P)dP/(P-f(P)) After separating the two integrals, I will try to solve them separately. For the first integral, I know that it is a standard integral and can be solved using the Fundamental Theorem of Calculus.For the second integral, I am not sure. It looks like it might involve the natural log of a Fourier transform, but I am not sure how to proceed.