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Unformatted text preview: Step II. We ﬁnd the superposition of these solution which satisﬁes the nonhomogeneous initial conditions by means of Fourier analysis.
To carry out Step I, we substitute u(x, t) = f (x)g (t) into the wave equation
(4.18) and obtain
f (x)g (t) = c2 f (x)g (t).
We separate variables, putting all the functions involving t on the left, all the
functions involving x on the right:
Once again, the left-hand side of this equation does not depend on x, while the
right-hand side does not depend on t, so neither side can depend upon either x
or t. Therefore the two sides must equal a constant λ, and our equation becomes
2 g (t)
which separates into two ordinary diﬀerential equations,
c2 g (t) or g (t) = λc2 g (t), (4.19) f (x)
f (x) or f (x) = λf (x). (4.20) and Just as in the case of the heat equation, the homogeneous boundary condition
u(0, t) = u(L, t) = 0 becomes
f (0)g (t) = f (L)g (t) = 0,
and assuming that g (t) is not identically zero, we obtain
f (0) = f (L) = 0.
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- Winter '14