# Find the particular solution to 413 which in addition

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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 L= d2 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.

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