908
Appendix: All Solutions to the Problems
where, the constant c 2 can be determined by the normal characteristic function namely
the isoperimetric condition of the problem, there is
1
0
⎧
⎨
⎩
c 2 sin
nπ ln(x+1)
ln 2
√
x + 1
⎫
⎬
⎭
2
dx =
1
2
c
2
2
1
0
1 − cos
nπ ln(x+1)
ln 2
x + 1
dx =
1
2
c
2
2 ln 2 = 1
(10)
Getting c 2 = ±
2
ln 2
. So the characteristic functions are
y n = ±
√
2 sin
nπ ln(x+1)
ln 2
√ (x + 1) ln 2
(n = 1, 2, . . .)
(11)
8.22 Find the characteristic value and characteristic function of the functional J [y] =
3
0 [(2x + 3)y
2
− y
2
]dx, such that it conforms to the boundary conditions y(0) =
y(3) = 0 and the isoperimetric condition
3
0 y
2 dx = 1.
Solution The Sturm-Liouville corresponding to the functional is
−y − 2(2x + 3)y
− (2x + 3)
2 y
= λy
(1)
or
(2x + 3)
2 y
+ 2(2x + 3)y
+ (λ + 1)y = 0
( 2 )
Making the substitution, 2x + 3 = e
T , then Eq. (2) can be transformed into the
second-order linear differential equation with constant coefficients
4
d
2 y
dt 2 + 4
dy
dt
+ (λ + 1)y = 0
( 3 )
The characteristic equation of the equation is
4r
2
+ 4r + λ + 1 = 0
( 4 )
The two roots of Eq. (44) are r 1,2 = −
1
2
±
1
2
√
−λ. When λ > 0, the general
solution of Eq. (3) is
y(t) = e
−
1
2 t
c 1 cos
√ λ
2
t + c 2 sin
√ λ
2
t
(5)
Thus the general solution of Eq. (2) is