# Predictor-corrector, starting values with Taylor method

1. Feb 23, 2012

### Wyrm

1. The problem statement, all variables and given/known data
Hi! I need to solve the following ODE:

$$xy'=1-y+x^2y^2, \qquad y(0)=1$$

using a predictor-corrector method. Starting values need to be found using a Taylor method.
The exact solution is of the form $$\frac{\tan{x}}{x}$$

2. Relevant equations

Taylor method of third order (for 1st starting value):

$$y_1=y_0+hy_0'+\frac{h^2}{2}y_o''+\frac{h^3}{6}y_o'''$$

3. The attempt at a solution
Right here I bumped into a problem: in order to evaluate the 1st starting value (for example, if I'm using Adams-Bashforth 3rd order as a predictor and Adams-Moulton 3rd order method as a corrector) I get the following:

$$y'=\frac{1}{x}-\frac{y}{x}+xy^2$$

Dividing with $$x=0$$ in order to get yields infinity :(. Any help would be appreciated.