ODE Change of Variable: Solving Separable Equations with u = y/x

[dipole]

Homework Statement

I have the ODE y' = f(\frac{y}{x}), and I want to re-write this as a separable equation using the change of variable u = \frac{y}{x}

The Attempt at a Solution

I use the chain rule to write y&#039; = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}<br /> = \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u)

which is a separable equation. However this seems to be wrong somehow because when I try using it to solve equations of the above form, I'm getting the wrong answer. Any help where I went wrong?

 

[jackmell]

Homework Statement

I have the ODE y&#039; = f(\frac{y}{x}), and I want to re-write this as a separable equation using the change of variable u = \frac{y}{x}

The Attempt at a Solution

I use the chain rule to write y&#039; = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}<br /> = \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u)

which is a separable equation. However this seems to be wrong somehow because when I try using it to solve equations of the above form, I'm getting the wrong answer. Any help where I went wrong?

If u=y/x then y&#039;=x\frac{du}{dx}+u

 

[dipole]

I don't see how I can use that to put the original equation \frac{dy}{dx} = f(\frac{y}{x}) into separable form though. :\

 

[jackmell]

Won't you then have the equation:

xu&#039;+u=f(u)

Ain't that separable?