# Math Help - Homogeneous differential equation...

1. ## Homogeneous differential equation...

Was wondering if anyone could help me out with this ODE:

(y-x)dy/dx + (2x+3y) = 0

I'm told that it's a homogeneous equation, and so presume that the substitution y=ux (i.e. u = y/x) will come into play somewhere. However when I try this it doesn't seem to help too much - is this substitution the right kind of approach or am I missing something?

Thanks

2. Let $y=ux$

$\therefore \frac{dy}{dx} = u + x\frac{du}{dx}$

then ur equation $\frac{dy}{dx}=\frac{2x+3y}{x-y}$ is,

$u + x\frac{du}{dx}=\frac{2+3u}{1-u}$

$\therefore x\frac{du}{dx}=\frac{2+3u}{1-u}-u$

$\therefore x\frac{du}{dx}=\frac{2+2u+u^2}{1-u}$

$\therefore \frac{(1-u)du}{2+2u+u^2}=\frac{dx}{x}$

Now integrate..