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Therefore, the solution of the eigenvalue problem (8.25)-(8.26) is an infinite sequence
of nonnegative simple eigenvalues and their associated eigenfunctions. We use the con-
venient notation:
πnx πn
2
X n (x) = cos , λ n = ( ) , n = 0, 1, 2, . . .
L L
Consider now the ODE (8.24) for λ = λ n . The solutions are

T 0 (t) = γ 0 + δ 0 t, (8.28)
p p
2
2
T n (t) = γ n cos( λ n c t) + δ n sin( λ n c t), n = 1, 2, 3, . . . (8.29)
Thus, the product solutions of the initial boundary value problem are given by
A 0 + B 0 t
u 0 (x, t) = X 0 (x)T 0 (t) = ,
2

πnx cπnt cπnt
u n (x, t) = X n (x)T n (t) = cos A n cos + B n sin , n = 1, 2, 3, . . .
L L L

Applying the (generalized) superposition principle, the expression
∞
A 0 + B 0 t X cπnt cπnt πnx
u(x, t) = + A n cos + B n sin cos (8.30)
2 L L L
n=1
is a (generalized, or at least formal) solution of the problem (8.16)-(8.19). It can be
proved that the solution (8.30) can be represented as a superposition of forward and
backward waves. In other words, solution (8.30) is also a generalized solution of the wave
equation.
It remains to find the coefficients A n , B n in solution (8.30). Here we use the initial
conditions. Assume that the initial data f, g can be expanded into generalized Fourier
series with respect to the sequence of the eigenfunctions of the problem, and that these
series are uniformly converging. That is,
∞
X πnx
a 0
f(x) = + a n cos , (8.31)
2 L
n=1
∞
X πnx
ea 0
g(x) = + ea n cos . (8.32)
2 L
n=1
Again, the (generalized) Fourier coefficients of f and g can easily be determined; for
m ≥ 0, we multiply (8.31) by the eigenfunction cos(πmx/L), and then we integrate over
[0, L]. We obtain
∞
Z L πmx Z L πmx X Z L πmx πnx
cos f(x)dx = a 0 cos dx + a n cos cos dx. (8.33)
0 L 2 0 L n=1 0 L L
It is easily checked that

0, m 6= n,
Z L πmx πnx 
cos cos dx = L/2, m = n 6= 0, (8.34)
0 L L 

L, m = n = 0.

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