Practical work #10.Picard’s method of successive approximations.

Practical work #10. Picard’s method of successive approximations.

Example 1. Given the differential eqn.

with the initial condition y = 0 when x = 0. Use Picard’s method to obtain y for x = 0.25, 0.5 and 1.0 correct to three decimal places.

Sol. (a) The given initial value problem is

where y = y 0 = 0 at x = x 0 = 0

We have first approximation,

(1)

Second approximation,

(2)

From (1) and (2), we see that y (1) and y (2) agree to the first term . To find the range of values of x so that the series with the term alone will give the result correct to three decimal places, we put

which gives,

Hence,

and

To find y (1.0), we make use of eqn. (2) which gives,

Example 2. Use Picard’s method to obtain y for x = 0.2. Given:

with initial condition y = 1 when x = 0.

Sol. Here f (x, y) = x – y, x 0 = 0, y 0 = 1

We have first approximation,

Second approximation,

Third approximation,

Fourth approximation,

Fifth approximation,

When x = 0.2, we get

Thus, y =.837 when x =.2.

Example 3. Approximate y and z by using Picard’s method for the particular solution of

given that y = 2, z = 1 when x = 0.

Sol. Let

Here,

We have,

Also,

First approximation,

and

Second approximation,

PROBLEM SET

1. Solve by Picard’s method, the differential equations where y = 1, at x = 0.

Obtain the values of y and z from III approximation when x = 0.2 and x = 0.5.

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