Approximate Methods for First Order Differential Equations

Example: First find the exact solution for the initial value problem

dy/dx = 1 + y², y(0) = 3 Note: The d.e. is nonlinear.

Strategy: Use separation of Variables:

Separate variables: dy / (1 + y² ) = dx

Integrate: ∫ dy / (1 + y² ) = ∫ dx which gives: tan^˗1 (y) = x + C

Or y = tan (x + C) Use the initial condition y(0) = 3 to obtain C = tan^-1(3)

The result is: y(x) = tan [ x + tan^-1(3) ]

Example: Use Picard's Method to find the first, second, and third approximations to

dy/dx = 1 + y², y(0) = 3 i.e. y₁(x), y₂(x), and y₃(x)

and evaluate estimate at x = 0.1. Further compare with the exact solution.

Strategy: Use successive approximations by applying the following:

x

y_n+1(x) = b + ∫ f [ t,y_n(t) ] dt

a

where b = 3 and a = 0 and start with y_o = b = 3

x x

So y₁(x) = 3 + ∫ [ 1 + y_o(t)² ] dt = 3 + ∫ [ 1 + 3² ] dt = 3 + 10 x

0 0

x x

Now y₂(x) = b + ∫ f [ t,y₁(t) ] dt = 3 + ∫ [ 1 + (3 + 10 t )² dt

a 0

So y₂(x) = 3 + 10 x + 30 x² + (100/3) x³

Click here to continue with this example.

Return to Differential Equations Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.