Homework 6, due October 20 Math 4041, Fall 2009
1. Let aand bbe positive numbers. Find all numbers λfor which there is a nonzero
function u: [0, a]×[0, b]→Rof the form u(x, y) = X(x)Y(y), with Xand Yboth
C2, such that
∂2u
∂x2+∂2u
∂y2−λu = 0 for (x, y)∈[0, a]×[0, b]
u(x, y) = 0 if y∈[0, b] and x= 0 or x=a
∂u
∂y (x, y) = 0 if x∈[0, a] and y= 0 or y=b
2. Let u1and u2be solutions of Problem 1 corresponding to two different numbers
λ1,λ2. Verify that
ZZ[0,a]×[0,b]
u1(x, y)u2(x, y)dx dy = 0
1
