Example 9.  Use Picard iteration to find the solution of the second order I.V.P.  

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Solution 9.

First define the function  , and   and the initial conditions        by typing:

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

We are done.

Techniques from calculus can be used to find the solution  .

We can express    as a Maclaurin series and observe that the sequence    is converging to the solution

For this example all the terms in the Picard approximation agree with all of the terms in the Maclaurin series.  

Aside.  We can let Mathematica solve the differential equation.  This is just for fun.

(c) John H. Mathews 2005