Verify that u n ( x , t ) given in equation ( 10 ) satisfies equation ( 1 ) and the boundary conditions in ( 2 ) by substituting u n ( x , t ) directly into the equations involved. u n ( x , t ) = X n ( x ) T n ( t ) = a n sin ( n π x / L ) b n e − β ( n π / L ) 2 t ( 10 ) ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 ( x , t ) , 0 < x < L , t > 0 ( 1 ) u ( 0 , t ) = u ( L , t ) = 0 , t > 0 , ( 2 )
Verify that u n ( x , t ) given in equation ( 10 ) satisfies equation ( 1 ) and the boundary conditions in ( 2 ) by substituting u n ( x , t ) directly into the equations involved. u n ( x , t ) = X n ( x ) T n ( t ) = a n sin ( n π x / L ) b n e − β ( n π / L ) 2 t ( 10 ) ∂ u ∂ t ( x , t ) = β ∂ 2 u ∂ x 2 ( x , t ) , 0 < x < L , t > 0 ( 1 ) u ( 0 , t ) = u ( L , t ) = 0 , t > 0 , ( 2 )
Solution Summary: The author explains how to prove that u_n(x,t) satisfies the given condition.
Verify that
u
n
(
x
,
t
)
given in equation (10) satisfies equation (1) and the boundary conditions in (2) by substituting
u
n
(
x
,
t
)
directly into the equations involved.
u
n
(
x
,
t
)
=
X
n
(
x
)
T
n
(
t
)
=
a
n
sin
(
n
π
x
/
L
)
b
n
e
−
β
(
n
π
/
L
)
2
t
(
10
)
∂
u
∂
t
(
x
,
t
)
=
β
∂
2
u
∂
x
2
(
x
,
t
)
,
0
<
x
<
L
,
t
>
0
(
1
)
u
(
0
,
t
)
=
u
(
L
,
t
)
=
0
,
t
>
0
,
(
2
)
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