# Math 257/316, Midterm 2, Section 101 17 November 2006

```Math 257/316, Midterm 2, Section 101
17 November 2006
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
Maximum score 100.
1. Let f (x) = cos(2x)
(a) Determine the sine series expansion for f (x) on the interval 0 &lt; x &lt; π.
(b) Is the Fourier series for f (x) on the interval −π &lt; x &lt; π the same expansion as you found above
in part (a)? Is there a way to answer this without working out the expansion?
Hint: It may be useful to know that
Zπ
sin(nx) cos(2x)dx =
0
&frac12;
0 if n is even
if n is odd
2n
n2 −4
[20 marks]
2. Solve the following inhomogeneous initial boundary value problem for the heat equation:
ut = uxx + e−t sin x, 0 &lt; x &lt; π, t &gt; 0
u(0, t) = 0 and u(π, t) = 0
u(x, 0) = 0
[40 marks]
3. Determine the solution to the following boundary value problem:
1
1
urr + ur + 2 uθθ
r
r
= 0, 0 &lt; r &lt; a, 0 &lt; θ &lt;
u(r, 0) = 0 = u(r,
π
2
r→0
π
), u(r, θ) &lt; ∞
2
u(a, θ) = sin(4θ)
[40 marks]
1
```