Page 117 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes In this way we can define y (x), y (x),...... successively in the domain |x x | < h. The functions
3 4 0
f (x), n = 1, 2, 3, ..... all regular in the domain |x x | < h and
n 0
|y (x) y | b.
n 0
So taking the integral along the segment connecting x and x we can prove that the sequence of
0
regular functions |y (x)| converges uniformly in the domain |x x | < h and that the limit
n 0
function y(x) satisfies
dy ( )
x
y(x ) = y and = f(x, y)
0 0 dx
in the domain |x x ) < h. As y(x) being the uniform limit of the sequence of regular functions is
0
also regular.
The Method of Undetermined Coefficients

Since in the previous section we have guaranteed the existence of the regular solution y(a), we
can calculate this solution by the method of undetermined coefficients as follows. By virtue of its
regularity, y(x) can be expanded in a power series
2
2
dy (x x 0 ) d y
y(x) = y + (x x ) ....
0 0 dx 2 dx 2
0 x
in the domain |x x | < h. Substituting this expansion for y on the right hand side of the
0
equation and differentiating we obtain
dy
= f(x, y)
dx
2
x
y
x
y
d y f ( , ) f  ( , ) dy
=
dx 2 x  y  dx
.......................................................
.......................................................
setting in these equations x = x and y = y we can determine successively the expansion coefficients
0 0
2
dy d y 3 y
, , ......
dx dx 2 x 3
0 x 0 x 0 x
6.5 Summary

 Picard method of finding the conditions under which the solution of the first order
differential equation is described.

 The method involves on the successive approximation and proving the uniform
convergence of the series. It also reduces to an integral equation.
 The Picard method of successive approximation does not find favour of the method of
existence as compared to Cauchy’s method of comparison test or other numerical methods
like Runge’s method.

110 LOVELY PROFESSIONAL UNIVERSITY