Let $D$ be the disk of radius 1 centered at (0,0). Find a formula for the solution of
$\Delta u=f$ in $D$
$u=1$ on $\partial D$
In the case where $f(x)=x_2$
In polar coordinates $f(r,\theta)=rsin(\theta)$
Supposed to use separation of variables to solve, but I guess I don't really understand how to do that.
When asked to solve $\Delta u = f$ with a polynomial function $f$, it's feasible to get some solution $p$ (without regard to boundary conditions) just by trying a polynomial of degree $\deg f+2$. What polynomial $p$ has $$ \frac{\partial^2 p}{\partial x_1^2}+\frac{\partial^2 p}{\partial x_2^2} = x_2\quad ? $$ ... I think a multiple of $x_2^3$ will do the job.
Then you are left with the task of finding a harmonic function with boundary value $x_2-p(x_1,x_2)$, which will not be hard if you use the identity (valid on the boundary) $$ x_2^3 = \sin^3\theta = \frac14(3\sin\theta-\sin(3\theta)) $$