Then the sequence
Method (3.4) is called the Picard iterative method and it is used to prove the existence of the solution of systems of ODE (3.1). It is mainly of theoretical value, as it should not necessarily be seen as a practical way to construct a numerical solution. However, it does give us insights into the qualitative properties of the solution. On the other hand, it is a useful exercise to construct the sequence of iterates in Equation (3.4) for some simple cases.
We note that the IVP (3.1) can be written as an integral equation as follows:
It can be proved that the solution of (3.1) is also the solution of (3.5) and vice versa. We see then that Picard iteration is based on (3.5) and that we wish to have the iterates converging to a solution of (3.5).
We note that the Picard method is used primarily to prove the existence of solutions and it is not a numerical method as such.
3.2.1 An Example
We take a simple autonomous non-linear scalar ODE to show how to calculate Picard iterates:
whose solution is given by:
We now compute the Picard iterates (3.4) for this ODE in order to determine the values of t for which the ODE has a solution. For convenience, let us take
(3.7)
We can see that the series is beginning to look like
3.3 OTHER MODEL EXAMPLES
We take some model ODEs for motivation.
3.3.1 Bernoulli ODE
The Bernoulli ODE is named after Jacob Bernoulli. It is special in the sense that it is a non-linear equation having an exact solution:
In the cases
(3.9)
3.3.2 Riccati ODE
