You didn't give careful statements of your boundary conditions but I would expect they are something like this:
u(r,0) = 0, 0 < r < R
u(R,θ) = 0, 0 < θ < α
u(r,α) = 50, 0 < r < R
In particular you aren't giving u(0,0) and the first and third conditions would disagree if you tried. So I don't see where keeping the r-μ
is ruled out. And you need it to get your second boundary condition as I have shown. Your separated equation for P(r) has a singular point at r =0.
You can verify that u(r,θ) = Asin(μθ)sinh(μln(r/R)) satisfies the DE and the homogeneous BC's. But, like I said, you're on your own from here.