a. Find the solution u( x, y) of Laplaces equation in the rectangle 0 < x < a, 0 < y < b, that satisfies the boundary conditions u(0, y) = 0, u(a, y) = 0, 0 < y < b, u( x, 0) = 0, u( x, b) = g( x), 0 < x < a. b. Find the solution if g( x) = x, 0 x a/2, a x, a/2 x a. G c. For a = 3 and b = 1, plot u versus x for several values of y and also plot u versus y for several values of x. (Use enough terms in the Fourier series to accurately approximate the nonhomogeneous boundary condition.) G d. Plot u versus both x and y in three dimensions. Also draw a contour plot showing several level curves of u( x, y) in the xyplane.

L1 - 3 2 1 2 3 2 − 2 2x(1 − x ) 3+ 3x (1 − x ) 3 ex. a) Simplify: 2 2 (1 − x ) 3 2 1/3 2 3 2 −2/3 2x(1 − x ) + 3x (1 − x ) b) Solve for x: 2 2/3 =0 (1 − x )