Consider the problem with uₓₓ + uᵧᵧ = 0, 0 < x < a, 0 < y < b,
with uᵧ(x, 0) = uᵧ(x, b) = u(a, y) = 0, and u(0y) = f(y).
Use separation of variables to find solutions for (a) f(y) = y/b and (b) f(y) = sin²(πy/b)
Hint : consider a trigonometric identity for the square of the sine function for part (b)