Euler Method

Differential Equations

The Euler method is a method for solving differential equations. It uses a first-order Taylor series to estimate the next step.

Derivation

We start with the Taylor series expansion of y(x) around x_n:

y (x_{n + 1}) = y (x_{n}) + (x_{n + 1} - x_{n}) y^{'} (x_{n}) + \frac{( x _{n + 1} - x _{n} ) ^{2}}{2} y^{''} (x_{n}) + O ((x_{n + 1} - x_{n})^{3})

Define h as the step size, where h = x_(n+1) - x_n. Substituting h simplifies the equation:

y (x_{n + 1}) = y (x_{n}) + h y^{'} (x_{n}) + \frac{h ^{2}}{2} y^{''} (x_{n}) + O (h^{3})

Since we are given the differential equation y'(x) = f(x, y), we replace y'(x_n) with f(x_n, y_n):

y_{n + 1} = y_{n} + h f (x_{n}, y_{n}) + \frac{h ^{2}}{2} y^{''} (x_{n}) + O (h^{3})

To obtain Euler’s method, we truncate the expansion by ignoring the higher-order terms:

y_{n + 1} \approx y_{n} + h f (x_{n}, y_{n})

This results in the final explicit Euler update formula:

y_{n + 1} = y_{n} + h f (x_{n}, y_{n})

While the method is written such that it is applicable to first order equations, it also can be used to solve higher order equations.