2nd order PDE using integration by parts

1. Oct 28, 2013

perishingtardi

1. The problem statement, all variables and given/known data
Find the general solution of the equation
$$(\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0,$$
where $\zeta$ and $\eta$ are independent variables.

2. Relevant equations

3. The attempt at a solution
I set $X = \partial u/\partial\eta$ so that $$(\zeta - \eta)^2 \frac{\partial X}{\partial\zeta}=0.$$ Then $$\int (\zeta - \eta)^2 \frac{\partial X}{\partial\zeta} \, d\zeta=f(\eta).$$ I used integration by parts to obtain
$$(\zeta - \eta)^2X - 2\int \zeta X \, d\zeta + 2\eta \int X\, d\zeta = f(\eta),$$ but I'm not sure if this is the correct method, or how to proceed.

2. Oct 28, 2013

dirk_mec1

Hint: what is
$$\frac{ \partial \zeta} { \partial\eta}$$

3. Oct 29, 2013

perishingtardi

its zero?? how does that help though?