Prove that the functions (a)
u
(
x
,
t
)
=
sin
π
x
cos
4
π
t
, (b)
u
(
x
,
t
)
=
e
−
x
−
2
t
(c)
u
(
x
,
t
)
=
ln
(
1
+
x
+
t
)
are solutions of the wave equation with me specified initial-boundary conditions:
a
.
{
u
t
t
=
16
u
x
x
u
(
x
,
0
)
=
sin
π
x
for 0
≤
x
≤
1
u
t
(
x
,
0
)
=
0
for 0
≤
x
≤
1
u
(
0
,
t
)
=
0
for 0
≤
t
≤
1
u
(
1
,
t
)
=
0
for 0
≤
t
≤
1
b
.
{
u
t
t
=
4
u
x
x
u
(
x
,
0
)
=
e
−
x
for 0
≤
x
≤
1
u
t
(
x
,
0
)
=
−
2
e
−
x
for 0
≤
x
≤
1
u
(
0
,
t
)
=
e
−
2
t
for 0
≤
t
≤
1
u
(
1
,
t
)
=
e
−
1
−
2
t
for 0
≤
t
≤
1
c
.
{
u
t
t
=
u
x
x
u
(
x
,
0
)
=
ln
(
1
+
x
)
for 0
≤
x
≤
1
u
t
(
x
,
0
)
=
1
/
(
1
+
x
)
for 0
≤
x
≤
1
u
(
0
,
t
)
=
ln
(
1
+
t
)
for 0
≤
t
≤
1
u
(
1
,
t
)
=
ln
(
2
+
t
)
for 0
≤
t
≤
1