Calculus/Euler's Method

Euler's Method is a method for estimating the value of a function based upon the values of that function's first derivative.

The general algorithm for finding a value of $y(x) \$ is:

$y_{n+1} = y_n + \Delta x_{step} \cdot f(x_n,y_n), \$

where f is y'(x). In other words, the new value, $y_{n+1}$, is the sum of the old value $y_n$ and the step size $\Delta x_{step}$ times the change, $f(x_n,y_n)$.

You can think of the algorithm as a person traveling with a map: Now I am standing here and based on these surroundings I go that way 1 km. Then, I check the map again and determine my direction again and go 1 km that way. I repeat this until I have finished my trip.

The Euler method is mostly used to solve differential equations of the form

$y' = f(x,y), y(x_0) = y_0. \$

Examples

A simple example is to solve the equation:

$y' = x + y, y(0) = 1. \$

This yields $f = y' = x + y$ and hence, the updating rule is:

$y_{n+1} = y_n + 0.1 (x_n + y_n)\$

Step size $\Delta x_{step}$ = 0.1 is used here.

The easiest way to keep track of the successive values generated by the algorithm is to draw a table with columns for $n, x_n, y_n, y_{n+1} \$.

The above equation can be e.g. a population model, where y is the population size and x a decease that is reducing the population.