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2nd order PDE using integration by parts

  1. Oct 28, 2013 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Oct 28, 2013 #2
    Hint: what is
    [tex] \frac{ \partial \zeta} { \partial\eta} [/tex]
     
  4. Oct 29, 2013 #3
    its zero?? how does that help though?
     
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