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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
The foundations of calculus of variations 13
problem via its Euler-Lagrange equation form. Note that the form of the in-
tegrand dictates the use of the extended form.
∂f
∂y
=0,
2
f
∂x∂y
=0,
2
f
∂y∂y
=0,
and
2
f
∂y
2
=
1
(1 + y
2
)
3/2
.
Substituting into the extended form gives
1
(1 + y
2
)
3/2
y

=0,
which simplifies into
y

=0.
Integrating twice, one obtains
y(x)=c
0
+ c
1
x,
clearly the equation of a line. Substituting into the boundary conditions we
obtain two equations,
y
0
= c
0
+ c
1
x
0
,
and
y
1
= c
0
+ c
1
x
1
.
The solution of the resulting linear system of equations is
c
0
= y
0
c
1
x
0
,
and
c
1
=
y
1
y
0
x
1
x
0
.
It is easy to reconcile that
y(x)=y
0
y
1
y
0
x
1
x
0
x
0
+
y
1
y
0
x
1
x
0
x
is identical to
y(x
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Publisher Resources

ISBN: 9781482253597