Example 6.  Use Picard iteration to find the solution to the system of differential equations

            

Solution 6.

First define the functions    and the initial conditions    by typing:

Picard iteration for generating the first six approximations is started with the Mathematica command:

    Techniques from calculus can be used to find the solutions      and   .  

The first few terms of these series can be shown to match the first few terms in the Picard iterations.

Mathematica can sum the infinite series to obtain the analytic solution.

We are done!    

Aside.  The following Mathematica commands will solve the system of D. E. 's.  

(c) John H. Mathews 2005