ENGINEERING MATHEMATICS II
Midterm 2 (May 5, 2021)
Instructor: Hao-Ming Hsiao
1. Consider the Sturm-Liouville system y" + λy = 0, y(−1) = y(1) = 0
(a) Find the eigenvalues,
(b) Determine the eigenfunctions,
(c) Expand f(x) = x for −1 ≤ 𝑥 ≤ 1 in a series of the found orthonormal
eigenfunctions.
(25%)
∞ cos pv
2. Given that ∫0
v2 +β2
dv =
𝜋
2𝛽
𝑒 −𝑝𝛽
(a) Find the Fourier transform of
1
.
x2 +b2
(10%)
(b) Use the result from (a) to solve for the integral equation y(x):
∞
y(u)
∫−∞ (x−u)2 +a2 du =
1
x2 +b2
(b) Use the result of (a) to evaluate
0<a<b
|𝑥| < 𝑎
|𝑥| > 𝑎
1
3. (a) Find the Fourier transform of f(x) = {
0

sin  a cos  x



(15%)
d
(10%)
(15%)
4. If ym(x) and yn(x) are two solutions of the Sturm-Liouville system
d 
dy 
p ( x)    q( x)   r(x)  y  0 (a < x < b) and both ym(x) and yn(x) are

dx 
dx 
continuous and differentiable, show that p(x) * W[ym(x), yn(x)] is a constant.
Note: W is the Wronskian.
(25%)