2nd order PDE using integration by parts

[perishingtardi]

Homework Statement

Find the general solution of the equation
[tex](\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0,[/tex]
where ##\zeta## and ##\eta## are independent variables.

Homework Equations

The Attempt at a Solution

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

 

[dirk_mec1]

Hint: what is
[tex] \frac{ \partial \zeta} { \partial\eta} [/tex]

 

[perishingtardi]

Hint: what is
[tex] \frac{ \partial \zeta} { \partial\eta} [/tex]

its zero?? how does that help though?