158
2 Variational Problems with Fixed Boundaries
Y = b 0 exp
a
1
2n x
+ b k exp
−a
1
2n x
+
n−1
k=1
(b k e
α k cos β k x + c k e
α k sin β k x) (7)
or
Y =
n
k=0
b k e
α k cos β k x +
n−1
k=1
c k e
α k sin β k x
(8)
When n is an even number, the characteristic Eq. (4) has no real root, has only n
pairs of conjugate complex roots, namely
r = a
1
2n
cos
(1 + 2k)π
2n
± i sin
(1 + 2k)π
2n
(k = 0, 1, · · · , n − 1)
(9)
Let
p k = a
1
2n cos
(1 + 2k)π
2n
, q k = a
1
2n sin
(1 + 2k)π
2n
(k = 1, 2, · · · , n − 1) (10)
then the solution of Eq. (3) is
Y =
n−1
k=0
(b k e
p k cos q k x + c k e
p k sin q k x)
(11)
Synthesize the particular solution y
∗ and the homogeneous solution Eq. (7) or
Eq. (11), the general solution of Eq. (2) is
y = b 0 exp
a
1
2n x
+ b k exp
−a
1
2n x
+
n−1
k=1
(b k e
α k cos β k x + c k e
α k sin β k x) + y
∗
(when n is odd number) (12)
y =
n−1
k=0
(b k e
p k cos q k x + c k e
p k sin q k x) + y
∗
(when n is even number)
(13)
where, the integral constants are determined by boundary conditions or the initial
conditions.