MA 221 Homework Solutions
Due date
:
April 2
/
3
,
2009
Sec 10.2 Problems #1
̄
9
̄
,2
̄
0
̄
̄
2
̄
Sec 10.6 Problem #3
̄
Section 10
.
2
Solve the vibrating string problem with
3,
L
, and the given initial functions
f
x
and
g
x
.
1
̄
9
̄
.)
f
x
3sin2
x
12sin13
x
,
g
x
0
By letting
3and
L
in formula (24) on page 585 of the text, we see that the
solution we want will have the form
u
x
,
t
∑
n
1
a
n
cos3
nt
b
n
sin3
nt
sin
nx
Therefore, we see that
∂
u
∂
t
∑
n
1
−
3
na
n
nt
3
nb
n
nt
sin
nx
In order for the solution to satisfy the initial conditions, we must find
a
n
and
b
n
such
that
u
x
,0
∑
n
1
a
n
sin
nx
x
x
,(
1
)
and
∂
u
x
∂
t
∑
n
1
3
nb
n
sin
nx
0.
From the first condition, we observe that we must have a term for
n
2,13 and for
these terms we want
a
2
a
13
12.
All of the other
a
n
’s must be zero. By comparing coefficients in the second condition,
we see that all
b
n
’s must also be zero.
Therefore, by substituting these values into equation (1) above, we obtain the solution
of the vibrating string problem with
3,
L
and
f
x
and
g
x
as given. This solution is given by
u
x
,
t
3
cos
3
2
t
sin2
x
12
cos
3
13
t
sin13
x
0
Or by simplifying, we obtain
u
x
,
t
3cos6
t
x
12cos39
t
x
2
̄
0
̄
.)
f
x
0,
g
x
−
2sin3
x
9sin7
x
−
sin10
x
By letting
L
in formula (24) on page 585 of the text, we see that the
solution we want will have the form
u
x
,
t
∑
n
1
a
n
nt
b
n
nt
sin