Example 1.  Use Picard iteration to find the solution of the I.V.P.  

        .  

Solution 1.

First define the function    and the initial condition    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 solution  .  

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    as a Maclaurin series and observe that the sequence    is converging to the solution.

         

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

(c) John H. Mathews 2005