
1158 SEPARATION OF VARIABLES AND INTEGRAL TRANSFORM METHODS
with the genera l initial condition (15.1.1.3) and the homogeneous boundary conditions
w = 0 at x = 0, w = 0 at x = l. (15. 1.2.6)
The func tion ψ(t) in the particular solution (15.1.1.5) is found from (15.1.2. 2), where γ(t) = 0:
ψ
n
(t) = exp(−λ
n
t). (15.1.2.7)
The functions ϕ
n
(x) are deter mined by solving the eigenvalue problem (15.1.1.6), (15.1.1.8) with
a(x) = 1, b(x) = c(x) = 0, s
1
= s
2
= 0, k
1
= k
2
= 1, x
1
= 0, and x
2
= l:
ϕ
′′
xx
+ λϕ = 0; ϕ = 0 at x = 0, ϕ = 0 at x = l.
So w e obtain the eigenfunctions and eigenvalues:
ϕ
n
(x) = sin
nπx
l
, λ
n
=
nπ
l
2
, n = 1, 2, . . . (15.1.2.8)
The solution of p r ...