Page 115 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes We obtain by assumption 2,
x x
|y(x) z(x)| = { ( , ( ) g ( , ( )}dt { ( , ( ) f ( , ( )}dt
t
t
z
f
t
z
t
t
t
z
f
t
y
t
0 x 0 x
x x
z
t
t
{ ( , ( ) g ( , ( )}dt K | ( ) z ( )|dt
f
t
t
t
y
z
t
0 x 0 x
x
t
t
y
 |x x | K | | |( ) z ( )dt
0 ...(5)
0 x
Therefore setting
x
y
SUP |x x 0 |  | ( ) z ( )| M ,
x
We have
|y(x) z(x)| |x x | + KM |x x |
0 0
for |x x | . Substituting this estimate for |y(t) z(t)| on the right hand side of (5), we obtain
0
M 'K 2 |x x | 2 2 K m 1 |x x | m
y
x
| ( ) z ( )| 0  0
x
2 m !
m 1
for |x x | . Repeating this substitution, we obtain, for each n = 1, 2, 3, ......,
0
M 'K n |x x | n n K m 1 |x x | m
x
y
x
| ( ) z ( )| 0  0
! n m !
m 1
for |x x | . As n the first term on the right hand side converges to zero uniformly on
0
the interval |x x | . The second term is less than
0
K
 K 1 {exp( |x x 0 |) 1}
Accordingly, the estimate of the error of the appropriate solution z(x) in the interval |x x |
0
is given by
|y(x) z(x)| ( K) (exp(K(x x ) 1) ...(6)
0
6.4 Solutions by Power Series Expansion
Consider the differential equation
dy
= f(x, y) ...(1)
dx
in the case when f(x, y) is a complex valued function of complex variables x and y. We assume that
,
f(x, y) can be expanded in a convergent power series in (x x ) and (y y ) in a domain D of the
0 0
complex (x, y) space given by
, ,
|x x | < a , |y y | < b .
0 0
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