Math 257/316, Midterm 2, Section 104
17 November 2008
Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.
Maximum score 100.
1. Solve the following inhomogeneous initial boundary value problem for the wave equation:
utt
= c2 uxx + e−t sin (5x) , 0 < x <
π
u(0, t) = 0 and ux ( , t) = t, t > 0
2
u(x, 0) = 0,
π
, t>0
2
ut (x, 0) = sin 3x + x, 0 < x <
π
2
[60 marks]
2. Use separation of variables to solve Laplace’s equation for the square plate:
uxx + uyy = 0, 0 < x, y < 1
ux (0, y) = 0 and ux (1, y) = 0
uy (x, 0) = 0, uy (x, 1) = cos(2πx)
Does this problem have a unique solution? If the above problem represents the steady state temperature
R1 R1
for a square plate whose average initial temperature was 100 degrees, i.e.
u(x, y, t = 0)dxdy = 100,
0 0
can you determine the solution uniquely?
[40 marks]
1