7.6 Variational Principles of Operator Equations
421
where, p(x), q(x) and f (x) are all the known functions of x.
Solution The differential equation can be rewritten as
( py
)
+ qy − f = 0
Making the inner product of the differential equation and δy in the interval [x 0 , x 1 ],
and performing integration by parts once, we get
x 1
x 0
[( py
)
+ qy − f ]δydx = py
δy
x 1
x 0
−
x 1
x 0
( py
δy
− qyδy + f δy)dx
= −
1
2
δ
x 1
x 0
( py
2
− qy
2
+ 2 f y)dx = 0
or
δ
x 1
x 0
( py
2
− qy
2
+ 2 f y)dx = 0
Thus the functional corresponding to the differential equation can be written by
J =
x 1
x 0
( py
2
− qy
2
+ 2 f y)dx
Whether or not the boundary conditions are given, the necessary boundary
condition is
py
δy
x 1
x 0
= 0
7.7 Variational Problems of Equivalence with Boundary
Value Problem of Self Conjugate Ordinary Differential
Equation
The differential equation like
n
k=0
(−1)
k d
k
dx k [ p k (x)y
(k)
] = f (x)
(7.7.1)
is called the self-conjugate ordinary differential equation, it is also called the
Sturm-Liouville 2 nth order differential equation. Where p k (x) ∈ C
k
[x 0 , x 1 ] and
p k (x) 0, p k (x) at most has only finite zero points. At this point Eq. (7.4.3) is a 2 nth