Picard iterative method to solve a differential equation

\[ \frac{dy}{dx} = f(x,y) , \qquad y(x_0)=y_0 \]

The Picard iterative process constucts a sequence of functions which will get closer and closer to the desired solution.

1. \[ y_0(x) = y_0 \]

2. the recurrent formula

\[ y_{n+1}(x) = y_0 + \int_{x_0}^x f(t,y_n(t))dt \,\qquad for n\ge 1\]

Example:

\[ y' = x(3-2y), \qquad  y(0)=1 \]

\[y_0 =1\]

\[y_1 = 1 + \int_0^x t(3-2*1) dt = 1 + x^2/2 \]

\[y_2 = 1+  \int_0^x t(3-2*y_1) dt = 1 + x^2/2 - x^4/4\]

\[ \ldots\]

\[\lim_{n\to \infty} y_n(x) = 3/2 -1/2 e^{-x^2} \]

Reference:

http://www.sosmath.com/diffeq/first/picard/picard.html

http://www.cfm.brown.edu/people/dobrush/am33/Mathematica/intro/picard.html