Appendix: All Solutions to the Problems
765
2b cos x − 2a sin x − x(a cos x + b sin x) + x(a cos x + b sin x) = 2 cos x
Comparing the coefficients of both sides, we get b = 1, a = 0. Therefore the
general solution of the differential equation is
y = Y + y
∗
= c 1 cos x + c 2 sin x + x sin x
From the boundary conditions y(0) = 0, y(π) = 0, we get c 1 = 0, c 2 = c is an
arbitrary constant, thus the extremal curve is
y = (c + x) sin x
2.10 Find the extremal curve of the functional J [y] =
π
0 (y
2
− 2y cos x)dx, the
boundary conditions are y(0) = y(π) = 0.
Solution Let the integrand F = y
2
− 2y cos x, the Euler equation of the functional
is
−2 cos x − 2y
= 0
Integrating two times, we get
y = cos x + c 1 x + c 2
From the boundary conditions y(0) = y(π) = 0, we get c 2 = −1, c 1 =
2
π
.
Therefore the extremal curve is
y = cos x +
2
π
x − 1
2.11 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 partial derivatives are
F y = 2y − 2 sin x, F y = 2y
, the Euler equation of the functional is
F y −
d
dx
F y = 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 = c 1 e
x
+ c 2 e
−x
+
1
2
sin x.