(Solved): Use this strategy to solve the Poisson equation with constant source and sinusoidal boundary data { ...

Use this strategy to solve the Poisson equation with constant source and sinusoidal boundary data $\{\begin{array}{rl}?u& =4,\\ u(1,?)& =\sin (2?).\end{array}$ Hint: For the particular solution to the non-homogeneous case, write the Laplacian in cartesian coordinates and try to think of the simplest function that has constant Laplacian (or use guess and check). Then write it in polar form. For the homogeneous case, recall that the general solution is a superposition of functions $u_k(r,?)=r^k\left(a_k\cos (k?)+b_k\sin (k?)\right)$ including the constant function when $k=0$. Also note that the data is "rigged" so you don't need to compute Fourier coefficients.