Improved Euler Method

Proof Technique
Consider the first order ODE:
 * $(1): \quad y' = f \left({x, y}\right)$ subject to the initial condition $y \left({x_0}\right) = y_0$

where $f \left({x, y}\right)$ is continuous.

Let $y \left({x}\right)$ be the solution of $(1)$.

For all $n \in \N_{>0}$, we define:
 * $x_n = x_{n-1} + h$

where $h \in \R_{>0}$.

Then for all $n \in \N_{>0}$ such that $x_n$ is in the domain of $y$:
 * $y_{n+1} = y_n + \dfrac h 2 \left({f \left({x_n, y_n}\right) + f \left({x_{n+1}, z_{n+1} }\right)}\right)$

where:
 * $z_{n+1} = y_n + h f \left({x_n, y_n}\right)$

is an approximation to $y \left({x_{n+1} }\right)$.