772
Appendix: All Solutions to the Problems
y = y 0 +
(y 1 − y 0 )[x
(1−n)
− x
(1−n)
0
]
x
(1−n)
1
− x
(1−n)
0
2.31 Find the extremal curve of the functional J [y] =
x 1
x 0
y
2
x k dx, where, k > 0.
Solution Let the integrand F =
y
2
x k , the integrand does not contain y, the first integral
of the Euler equation is
y
= cx
k
Integrating again, we get y = c 1 x
k+1
+ c 2 .
2.32 Find the extremal curve of the functional J [y] =
π
0 y
2 sin xdx.
Solution The Euler equation of the functional is
2y
cos x + 2y
sin x = 0
Separating the variables and integrating, we get
y
= c 1
1
sin x
Integrating once more, we get
y = c 1 ln
tan
x
2
+ c 2
The integral constants in the expression are determined by boundary conditions.
2.33 Find the extremal curve of the functional J [y] =
x 1
x 0
x
x+y dx.
Solution Because the integrand does not contain y, the Euler equation of the
functional has the first integral
x
(x + y ) 2 = c
or
y
= −x + c 0 x
1
2
Integrating we get
y = −
x
2
2
+ c 1 x
3
2 + c 2