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:

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 $$\Delta x_{\rm step}$$ 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

Examples
A simple example is to solve the equation:

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

Step size $$\Delta x_{\rm step}=0.1$$ 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 is time.