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Apply the initial conditions (8.7):
√
X(0) = C 1 = 0, X(l) = C 2 sin λx = 0,

but C 2 6= 0 implies
√
λl = πn.
The numbers
2 2
π n
λ n = , n = 1, 2. . . .
l 2
are eigenvalues of the problem (8.6),(8.7), and the functions

nπ
X n (x) = sin x , n = 1, 2, . . .
l
are the corresponding eigenfunctions. It is easy to see that the general solution of (8.5)
for λ = λ n has the form
√ √
T n (t) = A n cos λt + B n sin λt, n = 1, 2, . . .

Multiplying these solutions by the X n (x), corresponding to λ n , we find infinitely many
separated solutions

√ √ nπ
u n (x, t) = A n cos λt + B n sin λt sin x , for n = 1, 2, . . . ,
l
where A n , B n are arbitrary constants as before. Since a linear combination of solutions of
the wave equation is also a solution, any infinite sum

∞
X √ √ nπ
u n (x, t) = A n cos λt + B n sin λt sin x (8.8)
l
n=1
will also solve the wave equation.
It remains only to see if we can choose the A n ’s and B n ’s to satisfy

∞
X nπ
ϕ(x) = u(x, 0) = A n sin x (8.9)
l
n=1
∞
X πnc nπ
ψ(x) = u t (x, 0) = B n sin x (8.10)
l l
n=1
It is known that any (reasonably smooth) function defined on 0 < x < l has a unique
representation
∞
X nπ
h(x) = a n sin x , (8.11)
l
n=1
and we also know the formula
l
Z
2 nπ
c n = h(x) sin x dx
l l
0

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