Page 123 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes (4) Relations between the solution and the coefficients
Let y (x), y (x),...y (x) be a fundamental system of the solutions of (1). If every y (x) (i = 1, 2,
1 2 n i
.... n) satisfies another equation
n
d y d n 1 y i
r 1 .... r y
n i = 0
dx n dx n 1
With continuous coefficients r (x), i = 1, 2, ... n in the domain D then we have
i
n
x
r (x) p ( ), i 1, 2,... .
i i
This fact may be stated as follows:
The coefficients of a linear differential equation of the nth order are determined uniquely
by an arbitrary chosen fundamental system of the solutions, provided the coefficient of
n
d y
n is identically one.
dx
Let us write equation (1) as

y n p y n 1 p y n 2 ....p y = 0 ...(5)
i 2 n
with conditions

y
( y x 0 ) = , (x 0 ) ,....y (x 0 ) n ,....y n (x 0 ) n ...(6)
(5) Wronskian. Liouville s formula
We shall enter into the details of the relations between the solutions and the coefficients
mentioned above. We denote by W(y, y , y ,....y ) the determinant
1 2 n
y y 1 y 2 ... y n
y y y ... y
1 2 n
y y 1 y 2 ... y n
n
n
n
n
y ( ) y ( ) y ( ) ... y ( )
2
1
n
which is called the Wronskian of the n + 1 functions y, y , y , ...., y . We consider the linear
1 2 n
differential equation
x
x
y
W ( , y 1 ( ),y 2 ( ),....,y n ( )) = 0 ...(i)
x
where y is unknown and y 1 ( ),y 2 ( ),....,y n ( ) is a fundamental system of the solutions of
x
x
x
(5). Since
y
W ( ( ),y 1 ( ),y 2 ( ),...,y n ( )) = 0 (i = 1, 2, ...., n)
x
x
x
x
i
every y (x) satisfies the equation (i). Furthermore, as will be shown shortly, the coefficient
i
n
x
x
y
( 1) W ( ( ), y 2 ( ),..., y n ( )) ...(ii)
x
1
(n)
of y in (i) does not vanish at any point in the domain D. Therefore, we obtain the
following identity
n
x
y
( 1) W ( , y ( ),y ( ),...,y ( ))
x
x
n
x
x
y ( ) p ( )y (n 1) ... p ( )y p ( )y = 1 2 n ...(iii)
x
1 n 1 n
x
x
W ( ( ),y 2 ( ),...,y n ( ))
y
x
1
This gives the relations between the solutions and the coefficients.
116 LOVELY PROFESSIONAL UNIVERSITY