Example 5.  Use Picard iteration to find the solution to the system of differential equations

            

Solution 5.

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 analytic solutions      and   .  

Notice that when the last term in the Picard approximation is dropped, what is left is a Maclaurin (or Taylor) polynomial approximation.  

We can express the analytic solutions as Maclaurin series and observe that the sequences    and    are converging to the solution

           
      
         

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