I'm trying to understand a solved problem in our notes and can't find a similar technique used elsewhere, so I'm asking for some assistance in understanding what exactly they did.

The problem:

Find a general solution to the equation $\displaystyle x^2u_x + y^2u_y = (x + y)u$

and the particular solution when $\displaystyle u(x,1) = 1$

Solution:

The characteristic equations are:

$\displaystyle \frac{dx}{dt} = x^2, \frac{dy}{dt} = y^2, \frac{du}{dt} = (x + y)u$

I don't really understand these next steps they do which is:

These imply that for some constant $\displaystyle \alpha$

$\displaystyle \frac{x' - y'}{x - y} = \frac{u'}{u}, $ and $\displaystyle \frac{1}{x} - \frac{1}{y} = \alpha$

from which it follows that two integrals are for constants ($\displaystyle \beta, \gamma$)

$\displaystyle x - y = \beta u,$ or $\displaystyle u = \gamma xy$

There are a few more steps after this, but I guess I'll ask about those if I don't understand them after I understand these steps.

Thanks for any assistance.