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Unformatted text preview: f (x) or f (x) = λf (x). (4.10) and 86 The homogeneous boundary condition u(0, t) = u(L, t) = 0 becomes
f (0)g (t) = f (L)g (t) = 0.
If g (t) is not identically zero,
f (0) = f (L) = 0.
(If g (t) is identically zero, then so is u(x, t), and we obtain only the trivial
solution u ≡ 0.)
Thus to ﬁnd the nontrivial solutions to the homogeneous linear part of the
problem requires us to ﬁnd the nontrivial solutions to a boundary value problem
for an ordinary diﬀerential equation:
f (x) = d2
(f (x)) = λf (x),
dx2 f (0) = 0 = f (L). (4.11) We will call (4.11) the eigenvalue problem for the diﬀerential operator
dx2 acting on the space V0 of well-behaved functions f : [0, L] → R which vanish at
the endpoints 0 and L.
We need to consider three cases. (As it turns out, only one of these will
actually yield nontrivial solutions to our eigenvalue problem.)
Case 1: λ = 0. In this case, the eigenvalue problem (4.11) becomes
f (x) = 0, f (0) = 0 = f (L). The general solution to the diﬀerential equation is f (x) = ax + b, and the only
particular solution which satis...
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This document was uploaded on 01/12/2014.
- Winter '14