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 a continuous real function.

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

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

Proof
Let $(1)$ be integrated $x$ from $x_0$ to $x_1$.

Because $f$ is continuous, the assumption holds.

By making $h$ small, the difference:
 * $y_0 + h \, \map f {x_0, y_0} - \map f {x_1, y_1}$

can be made arbitrarily small.

$y_{n + 1}$ can be defined recursively:


 * EulerMethod.png

The errors accumulate; with increasing $n$ the values of $y_{n + 1}$ are based on increasingly inaccurate values of $y_n$.

These can be reduced by making $h$ smaller, so the inaccuracies can be reduced by increasing the computation needed.