For gx x2 0 lt x lt 1 write a generalized fourier series

Question

Math 054 Partial Differential Equations - HW Assignment 6

1. For g(x) = x², 0 < x < 1, write a generalized Fourier series in terms of the eigenfunctions

U_n(x) = √2 sin nπx, n = 1, 2, ...

2. Solve for u(x, y) (you may use results shown in class).

∇²u(x, y) = 0, 0 < x, y < 1

u_y(x, 0) = ½ - x, u_y(x, 1) = 0

u_x(0, y) = 0, u_x(1, y) = 0