Appendix: All Solutions to the Problems
845
The Euler equation of the functional is
2[q(x) + λr (x)]y −
d
dx
[2 p(x)y
] = 0
or
d
dx
[ p(x)y
] − [q(x) + λr (x)]y = 0
The boundary conditions are y(x 0 ) = 0, y(x 1 ) = 0. An obvious solution is y ≡ 0,
it does not satisfy the isoperimetric condition, however, other solutions exist only to
some values of λ, at the moment the value of λ is called the eigenvalue or characteristic value (see Chap. 8). Therefore, λ should be a characteristic value. A constant in
the general solution of the Euler equation is determined by the boundary condition
y(x 0 ) = 0, and another constant is determined by the isoperimetric condition.
5.8 Find the extremal function u = u(t) of the functional J =
t 1
t 0
u
2 dt, the fixed
boundary conditions are
x(0) = x 0 , x(t 1 ) = x 1 , y(0) = y 0 , y(t 1 ) = y 1
The constraint conditions are
x
= y, y
= ku
and find the constraint conditions and the extremal function under the following
two groups of cases
x(0) = x 0 , x(t 1 ) = y(0) = y(t 1 ) = 0
x(0) = y(0) = 0, x(1) = y(1) = 1
Solution 1 Making the auxiliary function H = u
2
+ λ 1 (t)(x
− y) + λ 2 (t)(y
− u).
The Euler equations of the functional are
H u = 2u − λ 2 = 0,
d
dt
H x = λ 1
= 0, H y −
d
dt
H y = λ 1 − λ 2
= 0
λ 1 = c 1 , λ 2 = c 1 t + c 2 , u =
c 1
2
t +
c 2
2
Integrating, we get
y =
c 1
4
t
2
+
c 2
2
t + c 3 , x =
c 1
12
t
3
+
c 2
4
t
2
+ c 3 t + c 4