MthSc 208 (Spring 2011)
Worksheet 7b
MthSc 208: Differential Equations (Spring 2011)
Inclass Worksheet 7b: The Wave Equation
NAME:
We will solve for the function
u
(
x, t
) defined for 0
≤
x
≤
π
and
t
≥
0 which satisfies the following initial
value problem of the wave equation:
u
tt
=
c
2
u
xx
u
(0
, t
) =
u
(
π, t
) = 0
,
u
(
x,
0) =
x
(
π

x
)
,
u
t
(
x,
0) = 1
.
(a) Carefully descsribe (and sketch) a physical situation that this models.
(b) Assume that
u
(
x, t
) =
f
(
x
)
g
(
t
). Compute
u
t
,
u
tt
,
u
x
,
u
xx
, and find boundary conditions for
f
(
x
).
(c) Plug
u
=
fg
back into the PDE and separate variables by dividing both sides of the equation by
c
2
fg
. Set this equal to a constant
λ
, and write down two ODEs: one for
f
(
x
) and one for
g
(
t
).
Written by
M. Macauley
1
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MthSc 208 (Spring 2011)
Worksheet 7b
(d) Solve the ODE for
f
(
x
) (including the boundary conditions), and determine
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 Spring '09
 Staufeneger
 Differential Equations, Equations, Boundary value problem, Partial differential equation, M. Macauley, Worksheet 7B, Inclass Worksheet 7b

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