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Unformatted text preview: b = 0 or sin(ωL) = 0, Now, however,
f (L) = 0 ⇒ and hence either b = 0 and we obtain only the trivial solution or sin(ωL) = 0.
The latter possibility will occur if ωL = nπ , or ω = (nπ/L), where n is an
integer. In this case, we obtain
f (x) = b sin(nπx/L).
Therefore, we conclude that the only nontrivial solutions to (4.11) are constant
multiples of
f (x) = sin(nπx/L), with λ = −(nπ/L)2 , n = 1, 2, 3, . . . . For each of these solutions, we need to ﬁnd a corresponding g (t) solving equation
(4.9),
g (t) = λc2 g (t),
where λ = −(nπ/L)2 . This is just the equation of exponential decay, and has
the general solution
2
g (t) = be−(ncπ/L) t ,
where a is a constant of integration. Thus we ﬁnd that the nontrivial product solutions to the heat equation together with the homogeneous boundary
condition u(0, t) = 0 = u(L, t) are constant multiples of
un (x, t) = sin(nπx/L)e−(ncπ/L) t .
2 88 It follows from the principal of superposition that
u(x, t) = b1 sin(πx/L)e−(cπ/L) t + b2 sin(2πx/L...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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