# Suppose that f1 x f2 x fn x are eigenfunctions

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Unformatted text preview: where h(θ) is a given function, periodic of period 2π , the initial temperature of the wire. Once again the heat equation itself and the periodicity condition are homogeneous and linear, so they must be dealt with ﬁrst. Once we have found the general solution to the homogeneous conditions ∂u ∂2u = c2 2 , ∂t ∂θ u(θ + 2π, t) = u(θ, t), we will be able to ﬁnd the particular solution which satisﬁes the initial condition u(θ, 0) = h(θ) by the theory of Fourier series. Thus we substitute u(θ, t) = f (θ)g (t) into the heat equation (4.23) to obtain f (θ)g (t) = c2 f (θ)g (t) and separate variables: g (t) f (θ) = c2 . g (t) f (θ) 103 The left-hand side of this equation does not depend on θ, while the right-hand side does not depend on t, so neither side can depend upon either θ or t, and we can write g (t) f (θ) = = λ, 2 g (t) c f (θ) where λ is a constant. We thus obtain two ordinary diﬀerential equations, g (t) = λ, c2 g (t) or g (t) = λc2 g (t), (4.24) f (θ) = λ, f (θ) or f...
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