Page 122 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Then by virtue of the Lipschitz condition, we obtain Notes

m x n
x
z m ,k ( ) z m ,k 1 ( ) K z m ,k 1 ( ) z m ,k 2 ( ) dt
t
t
x
n 1 0 x m 1
From this we obtain, for k > s
n k 1 (K x x ) t
x
x
z mk ( ) z m ,s ( ) nb 0 ...(10)
m 1 t s t
x
On the interval (9), provided that z m ,l ( ) y 0 m l . This suffices to prove the theorem.

7.2 General Properties of Solution of Linear Differential Equations
of Order n

We now discuss some of the properties of the solution of nth order linear differential equations.
For this purpose write down the differential equation in the form

d n d n 1 y
x
y p 1 ( ) 1 ... p y = p y q ( ) ...(1)
x
n
n
dx n dx n
The equation (1) is said to homogeneous if q(x) = 0, otherwise it is called inhomogeneous. We
,
x
assume that the coefficients p p 2 ,....p n , ( ) are all continuous on a domain D. We state that
q
1
(1) If y (x) and y (x) are any two non-zero solutions of equation (1) then y (x) + y (x) is also a
1 2 1 2
solution.
(2) In fact if y (x), y (x), y (x)...y (x) are solutions of equation (1) then any linear combination
1 2 3 n
m
y = c y i ...(2)
i
i 1
of these solutions with arbitrary coefficients c , c , ...., c is also a solution of (1). This fact
1 2 m
is called the principle of superposition.
(3) Let y (x), y ,...y be an arbitrary set of n + 1 solutions of equation (1), then there exist
1 2 n + 1
n + 1 numbers c , c , ...., c not all zero such that
1 2 n + 1
n 1
x
c y i ( ) = 0 ...(3)
i
i 1
that means that the set of n + 1 functions y , y, ...y is a dependent set.
1 n + 1
Thus if we have a set of n independent functions y ,.... y then the most general solution of
1 n
equation (1) is written as
n
y = c y i ...(4)
i
i 1
So a set of n solutions of y (x), y (x),...y (x), which are linearly independent is called a
1 2 n
fundamental system of the solutions of equation (1) (or general solution)

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