Indeed, often it is very hard to solve differential equations,
but we do have a numerical process that can approximate the
solution. This process is known as the **Picard iterative process**.

First, consider the IVP

It is not hard to see that the solution to this problem is also given as a solution to (called the integral associated equation)

The Picard iterative process consists of constructing a sequence of functions which will get closer and closer to the desired solution. This is how the process works:

**(1)**- for every
*x*; **(2)**- then the recurrent formula holds
for .

**Example:** Find the approximated sequence
, for the IVP

.

**Solution:** First let us write the associated
integral equation

Set . Then for any , we have the recurrent formula

We have , and

We leave it to the reader to show that

We recognize the Taylor polynomials of (which also get closer and closer to) the function

**
**

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