ABSTRACT
X′′(x) + λX(x) = 0,
X(0) = X(L) = 0
⎫ ⎬
⎭ (11.4)
has infinitely many solutions for the eigenvalues λn = (nπ L
)2 , n = 1, 2, . . ., and corresponding eigenfunctions Xn(x) = sin
) , n = 1, 2, ...; for all other values of λ, (11.4) has only
the trivial solution, that is, X(x) ≡ 0. We justified these conclusions in Example 9.14 in Section 9.3. Notice that the boundary conditions in (11.4) are the same as the boundary conditions
(11.2) applied to a function X(x) rather than to a function of two variables such as T(x, t).
