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Appendix: All Solutions to the Problems
2.12 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
− y
2
− 2y sin x)dx.
Solution Let the integrand F = y
2
− y
2
− 2y sin x, the Euler equation of the
functional is
2y − 2 sin x + 2y
= 0
or
y
+ y = sin x
This is the second order differential equation with constant coefficients, the general
solution is y = −
x cos x
2
+ c 1 cos x + c 2 sin x.
2.13 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
− y
2
− ay cosh x)dx,
where, a is a constant.
Solution Let the integrand F = y
2
− y
2
− ay cosh x, the Euler equation of the
functional is
2y − a cosh x + 2y
= 0
or
y
+ y =
a
2
cosh x
This is the second order differential equation with constant coefficients, the general
solution is y =
a
4
cosh x + c 1 cos x + c 2 sin x.
2.14 Find the extremal curve of the functional J [y] =
1
0 (y
2
+ 2ye
x
)dx, the
boundary conditions are y(0) = 0, y(1) = 1.
Solution The Euler equation of the functional is
2e
x
− 2y
= 0
Integrating two times, we obtain
y = e
x
+ c 1 x + c 2
From the boundary conditions y(0) = 0, y(1) = 1, we get c 1 = 2 − e, c 2 = −1,
thus the extremal curve is
y = e
x
+ (2 − e)x − 1