
62 Applied calculus of variations for engineers
Subtracting the equations and canceling the identical terms results in
(λ
2
− λ
1
)
D
u
1
u
2
dxdy =0.
Since
λ
1
= λ
2
,
it follows that
D
u
2
u
1
dxdy =0
must be true. The two eigensolutions are orthogonal. With similar arguments
and specially selected auxiliary functions, it is also easy to show that the sec-
ond solutions also satisfy
Δu
2
− λ
2
u
2
=0.
The subsequent eigensolutions may be found by the same procedure and
the sequence of the eigenpairs attain the extrema of the variational problem
under the successive conditions of the orthogonality against the preceding so-
lutions.
5.3 Sturm-Liouville problems
The process demonstrated ...