## 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).