# Math Help - Quasi linear PDE help

1. ## Quasi linear PDE help

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 $x^2u_x + y^2u_y = (x + y)u$
and the particular solution when $u(x,1) = 1$

Solution:
The characteristic equations are:
$\frac{dx}{dt} = x^2, \frac{dy}{dt} = y^2, \frac{du}{dt} = (x + y)u$

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

These imply that for some constant $\alpha$
$\frac{x' - y'}{x - y} = \frac{u'}{u},$ and $\frac{1}{x} - \frac{1}{y} = \alpha$

from which it follows that two integrals are for constants ( $\beta, \gamma$)
$x - y = \beta u,$ or $u = \gamma xy$

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.

2. From the first two, x'-y'=x^2-y^2=(x-y)(x+y) and using the third, x'-y'=(x-y)(u'/u)... Integrate by separating variables after that.