# 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 ${\displaystyle y(x)}$ is:

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

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

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

${\displaystyle y'=f(x,y),y(x_{0})=y_{0}}$

## Examples

A simple example is to solve the equation:

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

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

${\displaystyle y_{n+1}=y_{n}+0.1(x_{n}+y_{n})}$

Step size ${\displaystyle \Delta x_{\rm {step}}=0.1}$ 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 ${\displaystyle n,x_{n},y_{n},y_{n+1}}$ .

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