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Unformatted text preview: ons of the
u(x, t) = f (x)g (t).
By the superposition principle, an arbitrary linear superposition of these solutions will still be a solution.
Step II. We ﬁnd the particular solution which satisﬁes the nonhomogeneous
condition by Fourier analysis.
Let us ﬁrst carry out Step I. We substitute u(x, t) = f (x)g (t) into the heat
equation (4.8) and obtain
f (x)g (t) = c2 f (x)g (t).
Now we separate variables, putting all the functions involving t on the left, all
the functions involving x on the right:
The left-hand side of this equation does not depend on x, while the right-hand
side does not depend on t. Hence neither side can depend upon either x or t.
In other words, the two sides must equal a constant, which we denote by λ and
call the separating constant . Our equation now becomes
2 g (t)
which separates into two ordinary diﬀerential equations,
c2 g (t) or g (t) = λc2 g (t), (4.9) f (x)
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This document was uploaded on 01/12/2014.
- Winter '14