The equation ∂2u/∂x2 + ∂2u/∂y2 = u holds in the rectangle with vertices (0, 0), (a, 0), (a, b), (0, b). At the right vertical side, u(a, y) = sin(πy/b), 0 < y < b. The function u vanishes at the other three sides of the rectangle. Find u.
Since u vanishes at the two horizontal sides, we express u as a Fourier sine series of y :
