Math 316 Final Exam
April 20, 2010
Duration: 2 hours 30 minutes
First Name: Student Number: Last Name:
Do not open this test until instructed to do so. Relax.
This exam should have
13 pages, including this cover sheet. It is a closed book exam; no textbooks, calculators, laptops, formula sheets or other aids are allowed. Turn off any cell phones, pagers, etc. that could make noise during the exam. You must remain in this room until you have finished the exam.
Please explain your work, and circle your final solutions.
Use the extra pages if necessary.
Read these UBC rules governing examinations:
(i) Each candidate must be prepared to produce, upon request, a Library/AMS card for identification.
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(iv) Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action.
• Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other memory aid devices, other than those authorized by the examiners.
• Speaking or communicating with other candidates.
• Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
(v) Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
Problem Out of Score
1
2
10
10
3
4
20
20
5
6
Total
20
20
100
1

Problem 1 (10 points)
Consider the wave equation: u tt
= 4 u xx
, 0 < x < 8 , t > 0 , u (0 , t ) = 0 , u (8 , t ) = 0 ,
!
u ( x, 0) = f ( x ) =
1
0
3 ≤ x ≤ 5 otherwise.
, u t
( x, 0) = 0 .
In the coordinate systems provided below, carefully sketch the solution u ( x, t ) for t = 0, t = 1, t = 2, and t = 3.
a) t = 0 [2]
1 u(x,t)
0.5
0 1 2 3 4 5 6 7 8 x b) t = 1 [2]
1 u(x,t)
0.5
0 1 2 3 4 5 6 7 8 x
2

Problem 1 (continued) c) t = 2 [3]
1 u(x,t)
0.5
0 1 2 3 4 5 6 7 8 x d) t = 3 [3]
1 u(x,t)
0.5
0 1 2 3 4 5 6 7 8 x
3

Problem 2 (10 points)
Consider the following Laplace problem: u xx
+ u yy
= 0 , u (0 , y ) = 0 , u ( π, y ) = 0 , u ( x, 0) = 0 , u ( x, π ) = f ( x ) .
Find the general form of the solution u ( x, y ).
0 < x < π, 0 < y < π,
4

Extra page (Problem 2)
5

Problem 3 (20 points)
Consider the ordinary differential equation x
2 y !!
+ (2 x − x
2
) y !
+
"
3
2 x −
3
# y = 0 ,
4 x > 0 .
a) [3] Show that x = 0 is a regular singular point.
b) [10] By seeking solutions of the form y ( x ) =
$
∞ n =0 a n x n + r , find the roots of the indicial equation and find the recurrence relation for a n depending on r .
c) [7] For the larger of the two values of r , find the solution y ( x ) taking a
0
= 1.
6

Extra page (Problem 3)
7

Problem 4 (20 points) a) [5] Find the Fourier cosine coefficients for the function f ( x ) = x/π on 0 < x < π .
b) [15] Solve the steady heat conduction problem in the region of the half plane outside a semicircle: u rr
+
1 r u r
+
1 r 2 u
θθ
= 0 , r > 1 , 0 ≤ θ ≤ π, u
θ
( r, 0) = u
θ
( r, π ) = 0 , u (1 , θ ) =
θ
π
, u bounded as r → ∞ .
8

Extra page (Problem 4)
9

Problem 5 (20 points)
Consider the wave equation with periodic boundary conditions: u tt
= c 2 u xx
, − 1 < x < 1 , t > 0 , u ( − 1 , t ) = u (1 , t ) , u x
( − 1 , t ) = u x
(1 , t ) , u ( x, 0) = 1 + cos( πx ) , u t
( x, 0) = sin(2 πx ) .
Find the solution u ( x, t ).
10

Extra page (Problem 5)
11

Ignore this part
Problem 6 (20 points) a) [10] Consider the eigenvalue problem
X !!
+ µ
2
X = 0 , X !
(0) = 0 , X !
(1) = − X (1) .
Show that the eigenvalues satisfy µ tan µ = 1. Show graphically that there are infinitely many eigenvalues eigenfunctions X n
µ n and find their approximate value as n → ∞ . Find the
( x ) in terms of µ n
.
b) [10] Solve the heat conduction problem u t
= α 2 u xx
, 0 < x < 1 , t > 0 u x
(0 , t ) = 0 , u x
(1 , t ) = − u (1 , t ) , u ( x, 0) = 1 , giving your answer in terms of the eigenvalues µ n
.
You may use the result that a piecewise continuous function f ( x ) can be written as the generalized Fourier series f ( x ) = n =1 c n
X n
( x ) , where c n
=
&
1
0 & f ( x ) X n
0
1
X n
( x )
( x ) d x
2 d x
.
12

Extra page (Problem 6)
13