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Unformatted text preview: t, so both sides must be constant. If we let λ denote the constant, we obtain 1 c2 g (t) g (t) = λ = 1 f (x, y ) ∂2f ∂2f +2 2 ∂x ∂y . This separates into an ordinary diﬀerential equation g (t) = c2 λg (t), (5.9) and a partial diﬀerential equation ∂2f ∂2f +2 2 ∂x ∂y 124 = λf, (5.10) called the Helmholtz equation . The Dirichlet boundary condition yields f (0, y ) = f (a, y ) = f (x, 0) = f (x, b) = 0. In the second stage of separation of variables, we set f (x, y ) = X (x)Y (y ) and substitute into (5.10) to obtain X (x)Y (y ) + X (x)Y (y ) = λX (x)Y (y ), which yields X (x) Y (y ) + = λ, X (x) Y (y ) or X (x) Y (y ) =λ− . X (x) Y (y ) The left-hand side depends only on x, while the right-hand side depends only on y , so both sides must be constant, X (x) Y (y ) =µ=λ− . X (x) Y (y ) Hence the Helmholtz equation divides into two ordinary diﬀerential equations X (x) = µX (x), Y (y ) = νY (y ), where µ + ν = λ. The “Dirichlet boundary conditions” now become conditions on X (x) and Y (y ): X (0)...
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## This document was uploaded on 01/12/2014.

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