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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 onedimensional.
Moreover, the eigenvalues can be arranged in a sequence
0 > λ1 > λ2 > · · · > λ n > · · · ,
with λn → −∞. Finally, every wellbehaved 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.
 Winter '14
 Equations

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