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Unformatted text preview: ons of the special form 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: g (t) f (x) = c2 . g (t) f (x) 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 g (t) f (x) = = λ, 2 g (t) c f (x) which separates into two ordinary differential equations, g (t) = λ, c2 g (t) or g (t) = λc2 g (t), (4.9) f (x) = λ,...
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This document was uploaded on 01/12/2014.

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