Page 120 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Assuming that a 0, we can write the above equation in the form Notes
n
n
2
d y dy d y d n 1 y
= f x , , , ,..., ...(1)
y
dx n dx dx 2 dx n 1
We are interested in solving the equation (1) under the initial conditions
dy d n 1 y n 1
( y x 0 ) = y 0 , (x 0 ) y 0 ,...., n 1 (x 0 ) y 0 ...(2)
dx dx
Let us define

dy
= y
dx 1
dy
1 = y
dx 2
...........................
........................... (3)
...........................

dy n 2
dx = y n 1
dy n 1 f ( , ,y ,....,y )
x
y
dx = y n 1 n 1
with the initial conditions

( y x ) = y y (x ) y , y (x ) y ....y (x ) y (n 1) ...(4)
,
0 0 1 0 0 2 0 0 n 1 0 0
We may consider more generally, the system of ordinary differential equations
dz 1 f ( , , z ,.... )
x
z
z
dx = 1 1 2 n
dz 2 = f ( ,z z ,... )
z
,
x
dx 2 1 2 n
.................................................. (5)
..................................................
dz
n = f ( , z z ,....,z )
x
,
dx n 1 2 n
with the initial conditions
z m (x 0 ) = y 0 (m 1) , m 1, 2,...,n

where y (0) y . For this problem we shall prove the following theorem 1.
0 0
Theorem 1: Let
,
z
f 1 ( , z z 2 ,.... ), f 2 ( , z z 2 ,.... ),.... f n ( , z z 2 ,... ) ...(6)
,
z
x
x
,
x
z
n
1
n
1
1
n
)
,
be real valued and continuous on a Domain of the real ( ,x z z 2 ,...,z space given by
n
1
x x 0 , a z m y (m 1) , b m = 1, 2, ...n ....(7)
0
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