Page 114 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

dy
Unit 6: Existence Theorem for the Solution of the Equation = f(x, y)
dx

x 3 5 x 6 x 9 Notes
= 1 x
2 40 60 192
1 x 3 3 x 3 7 3
z = x x 4 x 5 x 8 dx
2 0 2 2 8 40 64
1 3 4 x 5 3 8 7 9 x 12
= x x x
2 8 10 64 360 256
and so on. So the series solution of y and z are convergent for x < 1.

Self-Assessment

1. Solve the differential equation
dy
= y
dx
under the initial conditions y = 1 for x = 1 by the method of successive approximations.
2. Solve the differential equation
dy
= x + y 2
dx
under the initial condition y = 0 when x = 0.

6.3 Remark on Approximate Solutions

On letting m in equation (15), we obtain

(K ) k
x
x
y ( ) y ( ) K
n ...(1)
k n
for |x x | . The equation (17) is an estimate of the error of the nth approximate solution
0
y (x). The method of successive approximation may be used, in principle. However this method
n
is not always practical because it requires one to repeat the evaluation of indefinite integrals
many times.
We shall now consider another method which is sometimes rather useful. Suppose that g(x, y) is
a suitable approximation to f(x, y) such that we can find the solution z(x) of the differential
equation

dz
= g(x, y) ...(2)
dx
On the interval |x x | satisfying the initial condition z(x ) = y . We put
0 0 0
y
f
x
y
x
SUP | ( , ) g ( , )|  ...(3)
x
( , ) D
y

Let y(x) be the unique solution of the differential equation
dy
= f(x, y) ...(4)
dx
on the interval |x x | h satisfying the initial condition y(x ) = y . Then from (2) it follows that
0 0 0
x
z
dt
t
t
f
y(x) z(x) = ( ( , ( )) g ( , ( )) .
t
t
y
0 x
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