Find
I'm guessing what you mean is find not . From your BC, it's desirable to switch to polar coords, so your PDE becomes
with your BC
We'll shift the BC a bit by introducing so the PDE is the same but the new BCs is . Fortunately for us, this problem admits separable solutions so
Applying the boundary condtions gives us (we can set wlog)
which leads to the final solution
Side note: Title says Legendre polynomial - not sure why? They typically arise in 3D geometries but nothing in this post suggest that the problem is in .