Math 2280:
PRACTICE PROBLEMS FOR THE FINAL TEST
This set of problems covers the last part of the course. Please refer to the practice midterm
tests for the rest of sample problems.
The test is closed book, closed notes test. Only small non-graphing calculator is allowed.
1. Solve the boundary value problem for a vibrating string of length π, the ends of the string
are held fixed, a = 2, if the initial shape of the string is given as sin 2πx and its initial
velocity is 1.
2. Solve the boundary value problem
∂2u
∂u
= 2 2,
∂t
∂x
0 < x < 1,
t > 0,
u(0, t) = 0,
u(1, t) = 0,
t > 0,
u(x, 0) = 3 sin(3πx) − 5 sin(5πx),
0<x<1
3. Find the Fourier cosine and sine series of the function f (x) and sketch the graphs of the
two extensions of f :
f (x) = sin(x), 0 ≤ x ≤ π
4. Find the inverse Laplace transform of F (s):
(a) F (s) = ln(s)
1
(b) F (s) = s(s−3)
5. Using Laplace transforms, solve the initial value problem:
x′′ + 4x = cos t,
x(0) = x′ (0) = 0.