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 $y' = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} = \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u)$

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 $y' = f(\frac{y}{x})$, 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 $y' = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} = \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u)$

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 $y'=x\frac{du}{dx}+u$

 

[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:

$$xu'+u=f(u)$$

Ain't that separable?