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Unformatted text preview: solutions to equation (5.5) are constant multiples of
X (x) = sin(nπx/a), with λ = −(nπ/a)2 , n = 1, 2, 3, . . . . For each of these solutions, we need to ﬁnd a corresponding Y (y ) solving
Y (y ) = −λY (y ),
where λ = −(nπ/a)2 , together with the boundary condition Y (0) = 0. The
diﬀerential equation has the general solution
Y (y ) = A cosh(nπy/a) + B sinh(nπy/a),
where A and B are constants of integration, and the boundary condition Y (0) =
0 implies that A = 0. Thus we ﬁnd that the nontrivial product solutions to
Laplace’s equation together with the homogeneous boundary conditions are constant multiples of
un (x, y ) = sin(nπx/a) sinh(nπy/a).
121 The general solution to Laplace’s equation with these boundary conditions is a
general superposition of these product solutions:
u(x, y ) = B1 sin(πx/a) sinh(πy/a) + B2 sin(2πx/a) sinh(2πy/a) + . . . . (5.7) To carry out Step II, we need to determine the constants B1 , B2 , . . . which
occur in (5.7) so that
u(x, b) = f (x).
Substitution of y = b into (5.7) yields
f (x) = B1 sin(πx/a) sinh(πb/a)...
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This document was uploaded on 01/12/2014.
- Winter '14