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

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

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:
 * $y_0 + h f \left({x_0, y_0}\right) - f \left({x_1, y_1}\right)$

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.