784
Appendix: All Solutions to the Problems
This is a second order linear differential equations, eliminating z, we get
y
(4)
+ 2y
+ y = 0
This is a fourth order differential equation with constant coefficients, it has a pair
of double complex root, r = ±i, thus it can work out y
y = (c 1 x + c 2 ) cos x + (c 3 x + c 4 ) sin x
Furthermore z = 2y + y
, from which z can be obtained
z = (c 1 x + c 2 ) cos x + (c 3 x + c 4 ) sin x − 2c 1 sin x + 2c 3 cos x
2.56 Find the extremal curve of the functional J [y, z] =
x 1
x 0
(y
2
+ z
2
+ y
z
)dx.
Solution Let the integrand F = y
2
+ z
2
+ y
z
, substituting it into the Euler
equations, we get
2y
+ z
= 0, 2z
+ y
= 0
or
y
= 0, z
= 0
Integrating we get two straight lines y = c 1 x + c 2 , z = c 3 x + c 4 .
2.57 Find the extremal curve of the functional J [y, z]
=
x 1
x 0
(y
2
+ z
2
− 2yz + 2y + 2z)dx.
Solution The Euler equations of the functional are
1 − z − y
= 0
1 − y − z
= 0
Deriving the first equation two times, then substituting the second equation into
the first equation, eliminating z
, we obtain
y
(4)
− y = 1
Similarly we obtain
z
(4)
− z = 1
The general solution is