Have you been told you need to use dz rd rd\theta ?
If so then rethink you co-ordinate system when integrating, and look at your functions and see if you can convert them into any other coordinate system you know of
I would begin by drawing a picture, perhaps several if you struggle thinking in 3D, and then try to work out what limits you're going to use in each direction.
Partial fractions won't work (as you've found), split it up into two bits and then use a table of Laplace transforms to work backwards, they are standard results - unless you're being made to there is no point in trying the integration yourself
Homework Statement
Find analytic solution of some kind:
0&=Y''(y)-\frac{\alpha^2 [u(y) + U]}{\epsilon}Y(y)
U eigenvalue, u(y) known, epsilon & alpha paramertres,and& Y thing to be found
Homework Equations
u(y) is a parallel flow of some kind and laplace transform given by...