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