Page 125 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Accordingly, W y 1 ( ),y 2 ( )..., y n ( ) transpose is a solution of the linear homogeneous
x
x
x
equation (vi) with coefficients continuous in D. Therefore, if W y 1 ( ),y 2 ( )..., y n ( )
x
x
x
x
x
vanishes at a point in D, then, W y 1 ( ),y 2 ( )..., y n ( ) is identically zero in the whole
x
domain D. This proves the following theorem.
Theorem 1: Either the Wronskian of n solutions of (5) is identically zero or it never vanishes
at any point in D.
By integration of the equation (vi), we obtain
x
x
t
W y ( ),y ( )..., y ( ) = W y (x ), y (x )..., y (x ) exp p ( )dt x D...(vii)
,
x
x
1 2 n 1 0 2 0 n 0 1
x 0
which is called Liouville s formula. From (3), it follows immediately that, if n solutions y (x),
1
y
x
x
x
y (x), .... y (x) of (5) are linearly dependent, then the Wronskian W ( ( ),y ( ),...,y ( )) is
2 n 1 2 n
identically zero on D. Thus we obtain the following:
Theorem 2: Let y (x), y (x), .... y (x) be n solutions of the equation (5). Then these solutions
1 2 n
x
are linearly independent if and only if the Wronskian W ( ( ),y 2 ( ),...,y n ( )) does not
y
x
x
1
vanish at any point in D. Further, these solutions are linearly dependent if and only if their
Wronskian is identically zero in D.
i
(6) Lagrange s method of variation of constants and D Alembert s method of reduction of
order
We shall be concerned with the inhomogeneous linear differential equation (1). Let y (x),
1
y (x) be solutions of (1). Then, clearly, y(x) = y (x) y (x) is a solution of the associated
2 1 2
homogeneous equation (5). This proves the following theorem.
Theorem 1: The general solution of (1) is written as the sum of a particular solution of (1)
and the general solution of (5).
However, if we know a fundamental system of the solutions of (5), then we can obtain a
particular solution of (1) by the method of variation of constants which is due to Lagrange.
Accordingly, in order to solve linear differential equations, it is sufficient to solve the
associated homogeneous equations.
The method of variation of constants. Let y , y , ..., y be a fundamental system of the solutions
1 2 n
of (5). Then the general solution of (5) is written in the form
n
x
y(x) = C y n ( ) ...(i)
i
i 1
Now we regard these constants C as functions of x, and try to determine them in such a
t
way that
n
x
y
y(x) = C t ( ) ( )
x
i
i 1
x
satisfies (1). As was shown by Lagrange, if C 1 ( ),C 2 ( ), ...,C n ( ) satisfy the system of
x
x
linear equations
x
x
x
x
x
x
y 1 ( )C 1 ( ) y 2 ( )C 2 ( ) ... y n ( )C n ( ) = 0
x
y ( )C ( ) y ( )C ( ) ... y ( )C ( ) = 0
x
x
x
x
x
1 1 2 2 n n
................................................................................. ...(ii)
y (n 2) ( )C ( ) y (n 2) ( )C ( ) ... y (n 2) C ( ) = 0
x
x
x
x
x
1 1 2 2 n n
118 LOVELY PROFESSIONAL UNIVERSITY