Chapter 5.3, Problem 9E

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

$y_{n+1}=y_n+\frac{h}{2}\left[f\left(x_n,y_n\right)+f\left(x_n+h,y_n+hf\left(x_n,y_n\right)\right)\right],n=0,1,\dots ,\mathrm{...}\left(9\right)$

where $x_{n+1}=x_n+h$. This explicit scheme is known as the improved Euler’s method.