Applied Calculus of Variations for Engineers, 2nd Edition

The inverse problem of calculus of variations 67

c

1

=



3

2

will enforce the satisfaction of



+1

−1

y

2

dx =1.

In fact, the generic form of the constants may be obtained by

c

i

=



2i +1

2

.

Finally, Sturm-Liouville problems may also be presented with functions of

multiple variables. For example, in three dimensions the equation becomes

−∇(p(x, y, z)∇u(x, y, z)) + q(x, y, z)u(x, y, z)=λu(x, y, z)

leading to various elliptic partial differential equations that all have engineer-

ing implications.