Prove that the functions (a)
u
(
x
,
t
)
=
sin
π
x
sin
2
π
t
, (b)
u
(
x
,
t
)
=
(
x
+
2
t
)
5
(c)
u
(
x
,
t
)
=
sin
x
cosh
2
t
are solutions of the wave equation with the specified initial-boundary conditions:
a
.
{
u
t
t
=
4
u
x
x
u
(
x
,
0
)
=
0
for 0
≤
x
≤
1
u
t
(
x
,
0
)
=
2
π
sin
π
x
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
)
=
x
5
for 0
≤
x
≤
1
u
t
(
x
,
0
)
=
10
x
4
for 0
≤
x
≤
1
u
(
0
,
t
)
=
32
t
5
for 0
≤
t
≤
1
u
(
1
,
t
)
=
(
1
+
2
t
)
5
for 0
≤
t
≤
1
c
.
{
u
t
t
=
4
u
x
x
u
(
x
,
0
)
=
sinh
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
)
=
1
2
(
e
−
1
e
)
cosh2
t
for 0
≤
t
≤
1