2.7 Variational Problems Depending on Higher Order Derivatives
157
Example 2.7.9 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
+ yy
)dx,
the boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 , y
(x 0 ) = y
0 , y
(x 1 ) = y
1 .
Solution Owing to J [y] =
x 1
x 0
(y
2
+ yy
)dx =
x 1
x 0
d(yy
), namely the integrand
is the total differential of the function yy
under the integral sign, the variational
problem is meaningless.
Example 2.7.10 Find the extremal function of the functional J [y] =
x 1
x 0
{a
−1 y
(n)2
+ [ f (x) − y]
2
}dx, where, f (x) is the known function of x.
Solution The Euler equation of the functional is
−[ f (x) − y] + (−1)
n a
−1 y
(2n)
= 0
( 1 )
or
y
(2n)
+ (−1)
n ay = (−1)
n a f (x)
(2)
This is a nonhomogeneous 2n th order linear differential equation with constant
coefficient.
By the theory of differential equations, the general solution of a nonhomogeneous
2n th order linear differential equation with constant coefficients is the sum of the
special solution of the nonhomogeneous equation and the solution of corresponding
homogeneous equation. For the right side of Eq. (2), according to the specific situation
of f (x), the special solution y
∗ can be solved.
The homogeneous equation on the left side of Eq. (2) is
Y
(2n)
+ (−1)
n aY = 0
( 3 )
Equation (3) is the 2n th order linear differential equation with constant coefficient,
the characteristic equation is
r
2n
− (−1)
n a = 0
( 4 )
When n is an odd number, the characteristic Eq. (4) has two real roots and n − 1
pairs of conjugate complex roots, namely
r = a
1
2n , −a
1
2n , a
1
2n
cos
2kπ
2n
± i sin
2kπ
2n
(k = 1, 2, · · · , n − 1)
(5)
Let
α k = a
1
2n cos
2kπ
2n
, β k = a
1
2n sin
2kπ
2n
(k = 1, 2, · · · , n − 1)
(6)
then the solution of Eq. (3) is
157
Example 2.7.9 Find the extremal curve of the functional J [y] =
x 1
x 0
(y
2
+ yy
)dx,
the boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 , y
(x 0 ) = y
0 , y
(x 1 ) = y
1 .
Solution Owing to J [y] =
x 1
x 0
(y
2
+ yy
)dx =
x 1
x 0
d(yy
), namely the integrand
is the total differential of the function yy
under the integral sign, the variational
problem is meaningless.
Example 2.7.10 Find the extremal function of the functional J [y] =
x 1
x 0
{a
−1 y
(n)2
+ [ f (x) − y]
2
}dx, where, f (x) is the known function of x.
Solution The Euler equation of the functional is
−[ f (x) − y] + (−1)
n a
−1 y
(2n)
= 0
( 1 )
or
y
(2n)
+ (−1)
n ay = (−1)
n a f (x)
(2)
This is a nonhomogeneous 2n th order linear differential equation with constant
coefficient.
By the theory of differential equations, the general solution of a nonhomogeneous
2n th order linear differential equation with constant coefficients is the sum of the
special solution of the nonhomogeneous equation and the solution of corresponding
homogeneous equation. For the right side of Eq. (2), according to the specific situation
of f (x), the special solution y
∗ can be solved.
The homogeneous equation on the left side of Eq. (2) is
Y
(2n)
+ (−1)
n aY = 0
( 3 )
Equation (3) is the 2n th order linear differential equation with constant coefficient,
the characteristic equation is
r
2n
− (−1)
n a = 0
( 4 )
When n is an odd number, the characteristic Eq. (4) has two real roots and n − 1
pairs of conjugate complex roots, namely
r = a
1
2n , −a
1
2n , a
1
2n
cos
2kπ
2n
± i sin
2kπ
2n
(k = 1, 2, · · · , n − 1)
(5)
Let
α k = a
1
2n cos
2kπ
2n
, β k = a
1
2n sin
2kπ
2n
(k = 1, 2, · · · , n − 1)
(6)
then the solution of Eq. (3) is