Math 257/316, Midterm 1, Section 104
20 October 2008
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
Maximum score 100.
1. Let f (x) = x on the interval 0 < x < π.
(a) Sketch the even and odd periodic extension of f with period 2π.
[5 marks]
(b) Use the half-range Fourier cosine series expansion to derive a series expansion for |x| on −π <
x < π. Now use this series to write down a series expansion for
½
x + π2 if − π < x < 0
g(x) =
−x + π2 if 0 ≤ x ≤ π
[15 marks]
(c) Use the series expansion for |x| to derive a series expansion for
2
π
8
.
[10 marks]
Hint: It may be useful to know that
Zπ
x cos(nx)dx =
0
½
0 if n is even
if n is odd
−2
n2
2. Apply the method of separation of variables to determine the solution to the one dimensional heat
equation with the following homogeneous boundary conditions (you need not give all the details of the
different cases for the eigenvalue problem):
∂u
∂t
=
BC
:
IC
:
∂2u
, 0 < x < π, t > 0
∂x2
∂u(0, t)
∂u(π, t)
=0=
∂x
∂x
u(x, 0) = 2 sin2 (x)
[20 marks]
3. Consider the second order differential equation:
Ly = 2x2 y 00 + 3xy 0 − (1 + x2 )y = 0
(1)
(a) Classify the points x ≥ 0 (do not include the point at infinity) as ordinary points, regular singular
points, or irregular singular points.
[5 marks]
(b) Use the appropriate series expansion about the point x = 0 to determine two independent solutions
to (1). You only need to determine the first three non-zero terms in each case.
[35 marks]
1