# Euler's Method

In mathematics, the Euler method is used to solve ordinary differential equations with a given initial value by numerical methods. Leonhard Euler has proved this method as the most basic explicit method and so the method is named after him. The Euler method is called as a first-order method, which means that the error per step(or local error) is proportional to the square of the step size, and the error at a given time (or global error) is proportional to the step size. Euler’s method also has stability problems. For all these reasons, the Euler method is not often used in practice but serves as the basis to construct more complicated methods.

To use Euler's Method to get a numerical solution to a problem of the form y' = f(x, y) with an initial value y(x$_o$) = y$_o$ we first decide on what interval, starting at the initial condition(x$_0$, y$_0$), we want to find the solution. We divide this interval into smaller intervals of length h.

Then, with the initial condition as the starting point, the rest of the solution is generated by using the iterative formulas:

x$_{n + 1}$ = x$_n$ + h

and y$_{n + 1}$ = y$_n$ + h f(x$_n$, y$_n$)Using these iterative formulas we can find the coordinates of the points in our numerical solution. We end this process when we arrive at the right end of the desired interval.