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Unformatted text preview: (x) f (x)g (t) = f (x)g (t), or = . ρ(x) g (t) ρ(x) f (x) 111 The two sides must equal a constant, denoted by λ, and the partial diﬀerential equation separates into two ordinary diﬀerential equations, T f (x) = λf (x), ρ(x) g (t) = λg (t). The Dirichlet boundary conditions (4.35) yield f (0) = 0 = f (1). Thus f must satisfy the eigenvalue problem L(f ) = λf, f (0) = 0 = f (1), where L= T d2 . ρ(x) dx2 Although the names of the functions appearing in L are a little diﬀerent than those used in the previous section, the same Theorem applies. Thus the eigenvalues of L are negative real numbers and each eigenspace is one-dimensional. Moreover, the eigenvalues can be arranged in a sequence 0 &gt; λ1 &gt; λ2 &gt; · · · &gt; λ n &gt; · · · , with λn → −∞. Finally, every well-behaved function can be represented on [a, b] as a convergent sum of eigenfunctions. If f1 (x), f2 (x), . . . , fn (x), . . . are eigenfunctions corresponding to the eigenvalues λ1 , λ2 , . . . , λn , . . . . Then the general solution to (4.34)...
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## This document was uploaded on 01/12/2014.

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