MthSc 208, Fall 2011 (Differential Equations)
Dr. Macauley
HW 19
Due Friday April 22nd, 2011
(1) Consider the ODE
y
00
= 4
y
. We know that the general solution is
y
(
t
) =
C
1
e
2
t
+
C
2
e

2
t
,
i.e.,
{
e
2
t
, e

2
t
}
is a
basis
for the solution space. Use the fact that
e
2
t
= cosh 2
t
+ sinh 2
t
and
e

2
t
= cosh 2
t

sinh 2
t
, and that any linear combination of solutions is a solution,
to find two distinct solutions involving hyperbolic sines and cosines.
Write the general
solution using these functions.
(2) We will solve for the function
u
(
x, t
), defined for 0
≤
x
≤
π
and
t
≥
0, which satisfies the
following conditions:
∂u
∂t
=
c
2
∂
2
u
∂x
2
,
u
(0
, t
) =
u
(
π, t
) = 0
,
u
(
x,
0) = 5 sin
x
+ 3 sin 2
x.
(a) Briefly describe, and sketch, a physical situation which this models. Be sure to explain
the effect of both boundary conditions (called
Dirichlet
boundary conditions) and the
initial condition.
(b) Assume that
u
(
x, t
) =
f
(
x
)
g
(
t
).
Find
u
t
and
u
xx
.
Also, determine the boundary
conditions for
f
(
x
) (at
x
= 0 and
x
=
π
) from the boundary conditions for
u
(
x, t
).
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 Spring '09
 Staufeneger
 Differential Equations, Equations, Boundary value problem, Boundary conditions, value problem, Dr. Macauley HW

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