Improved Euler Method

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

where $\map f {x, y}$ is continuous.

Let $\map y x$ 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 $y$:
 * $y_{n + 1} = y_n + \dfrac h 2 \paren {\map f {x_n, y_n} + \map f {x_{n + 1}, z_{n + 1} } }$

where:
 * $z_{n + 1} = y_n + h \map f {x_n, y_n}$

is an approximation to $\map y {x_{n + 1} }$.