Page 126 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 7: General Properties of Solutions of Linear Differential Equations of Order n

(n 1) (n 1) (n 1) Notes
x
x
x
x
y 1 ( )C 1 ( ) y 2 ( )C 2 ( ) ... y n C n ( ) = q(x)
x
x
x
y
then n C ( ) ( ) satisfies (1).
i 1 l i
In fact, if there exist C (x), C (x), ..., C (x) satisfying (ii), then, by differentiation and by
1 2 n
making use of (ii), we obtain successively
n
y
y(x) = C i ( ) ( )
x
x
i
i 1
n
x
x
y
y ( ) = C i ( ) ( )
x
i
i 1
..............................................
n
x
x
x
y (n 1) ( ) = C i ( )y (n 1) ( )
i
i 1
n
n
n
x
x
x
y ( ) ( ) = C i ( )y ( ) ( ) q ( )
x
i
i 1
Since y (x) satisfies (5), y(x) is certainly a solution of (1).
i
Now we consider the system (ii). According to Theorem 2, the Wronskian W(y (x), y (x), ...,
1 2
y (x)) of the fundamental system {y (x)} never vanishes at any point in the domain D, in
n i
which the coefficients p (x), p (x), ..., p (x) of (5) are continuous. Therefore, there exists one
1 2 n
x
and only one set of solutions C ( ),C ( ),...,C ( ) of (ii), which is written as
x
x
i 2 n
x
y
x
x
x
x
dC i ( )/dx = q ( )W i ( )/W ( ( ),y 2 ( ),....,y n ( )) ...(iii)
x
i
n
= Z i ( ), (i 1,2,..., )
x
x
where W (x) is the cofactor of y (n 1) ( ) in W(y (x), y (x),..., y (x)). Integrating (iii), we
i i 1 2 n
obtain
x
n
t
C (x) = Z i ( )dt C t , (i 1,2,..., ) ...(iv)
i x 0
where C is a constant of integration. Consequently, a particular solution of the equation
t
(1) is
n
x
t
y(x) = Z i ( )dt C y i ( ) ...(v)
x
i
x 0
x
The method of reduction of order. If a particular solution y (x), not identically zero, of the nth
1
order linear differential equation (5) is known, then, by setting
y = y z
1
(5) can be reduced to a linear differential equation of the (n 1) order with respect to dz/
dx. This procedure is called the method of reduction of order and is due to D Alembert.
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