MTHSC 208 (Differential Equations)
Dr. Matthew Macauley
HW 24
Due Friday April 24th, 2009
(1) Which of the following functions are harmonic?
(a)
f
(
x
) = 10

3
x
.
(b)
f
(
x, y
) =
x
2
+
y
2
.
(c)
f
(
x, y
) =
x
2

y
2
.
(d)
f
(
x, y
) =
e
x
cos
y
.
(e)
f
(
x, y
) =
x
3

3
xy
2
.
(2)
(a) Solve the following Dirichlet problem for Laplace’s equation in a square region: Find
u
(
x, y
), 0
≤
x
≤
π
, 0
≤
y
≤
π
such that
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0
,
u
(0
, y
) =
u
(
π, y
) = 0
,
u
(
x,
0) = 0
,
u
(
x, π
) =
x
(
π

x
)
.
(b) Solve the following Dirichlet problem for Laplace’s equation in the same square region:
Find
u
(
x, y
), 0
≤
x
≤
π
, 0
≤
y
≤
π
such that
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0
,
u
(0
, y
) = 0
,
u
(
π, y
) =
y
(
π

y
)
,
u
(
x,
0) =
u
(
x, π
) = 0
(c) By adding the solutions to parts (a) and (b) togeter, find the solution to the Dirichlet
problem: Find
u
(
x, y
), 0
≤
x
≤
π
, 0
≤
y
≤
π
such that
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0
,
u
(0
, y
) = 0
,
u
(
π, y
) =
y
(
π

y
)
,
u
(
x,
0) = 0
,
u
(
x, π
) =
x
(
π

x
)
.
(d) Sketch the solutions to (a), (b), and (c).
Hint: it is enough to sketch the boundaries,
and then use the fact that the solutions are harmonic functions
.