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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 lefthand side of this equation does not depend on y , while the righthand
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.
 Winter '14
 Equations

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