# and n 1 2 3 where bmn is a constant 532

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Unformatted text preview: g all the functions involving x on the left, all the functions involving y on the right: X (x) Y (y ) =− . X (x) Y (y ) The left-hand side of this equation does not depend on y , while the right-hand side does not depend on x. Hence neither side can depend upon either x or y . 120 In other words, the two sides must equal a constant, which we denote by λ, and call the separating constant, as before. Our equation now becomes X (x) Y (y ) =− = λ, X (x) Y (y ) which separates into two ordinary diﬀerential equations, X (x) = λX (x), (5.5) Y (y ) = −λY (y ). (5.6) and The homogeneous boundary condition u(0, y ) = u(a, y ) = 0 imply that X (0)Y (y ) = X (a)Y (y ) = 0. If Y (y ) is not identically zero, X (0) = X (a) = 0. Thus we need to ﬁnd the nontrivial solutions to a boundary value problem for an ordinary diﬀerential equation: X (x) = d2 (X (x)) = λX (x), dx2 X (0) = 0 = X (a), which we recognize as the eigenvalue problem for the diﬀerential operator L= d2 dx2 acting on the space V0 of functions which vanish at 0 and a. We have seen before that the only nontrivial...
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## This document was uploaded on 01/12/2014.

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