Laplace equation with boundary conditions in polar coordinates

Question

Show that the problem with this boundary conditions

$u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$

$u(r,0)=0$

$u(r,\pi) =T_0$

$u(1,\theta) =T_0 $

Have the solution $u(r,\theta)= T_0/\pi($$\theta+2 \sum_{n=1}^{\infty} r^n/n \cdot sen(\theta)$)

I don´t know what to do, I made the problem in two parts but it's not correct, any help will be apprecciated, thank you

Answer 1

Hint: Plug $u(r,\theta)$ into the PDE and check if it is a solution to to PDE. Also check your boundary values.