Question involving the solution to a Lagrange Differential Equation

[jbowers9]

Homework Statement

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

Let y' = P
taking d/dx and rearranging gives

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

a 1st order linear differential equation in standard form.

Homework Equations

When I attempt to solve by the suggested standard method, I end up with the following integral:

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

The Attempt at a Solution

I'm at a loss as how to go about integrating it.

 

[gabbagabbahey]

Try the substitution $u(p)=p-f(p)$

 

[jbowers9]

I did and I got

$$\int dP/u(P) - ln|u(P)|$$
What do I do with the 1st term?

 

[gabbagabbahey]

I don't think you can simplify it any further without knowing what $f$ is. At least your integral no longer involves $f'$ though.

 

[jbowers9]

P = y' ; y' = dy/dx I believe.

The following site lays it out: http://www.newcircuits.com/articles.php
under Lagrange differential equation.

I was lead to this site after setting up the Euler-Lagrange equations in r,phi,theta, and became curious as to how to solve them or what kind of solutions they have. ie exact, series. Thanks.