Classical Runge-Kutta 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 particular solution of $(1)$.

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

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

Let the following numbers be calculated:

Then for all $n \in \N_{>0}$ such that $x_n$ is in the domain of $y$:
 * $y_{n + 1} = y_n + \dfrac 1 6 \paren {m_1 + 2 m_2 + 2 m_3 + m_4}$

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

Also known as
This method is just one member of the family of Runge-Kutta Methods and so is often referred to with more specific names:
 * RK4
 * Classical Runge-Kutta Method

but usually it is simply known as the Runge-Kutta method.