## Euler's Method for Solving First Order Differential Equations

Euler's method allows us to find approximates values forat a point given the equationandSometimes this equation cannot be solved analytically, and Euler's method is a quick method to find values ofnumerically. We start by defining a step sizeWe assumeis constant on the intervalthenhence

Givenand a step sizewe can find

etc.

Example: Givenwithand a step size offind

For this simple example we can find the exact answer.

From this we findan error of less than 1%.

If the step size is reduced we would expect a more accurate estimate ofIn general if the step size is halved the error is reduced by a factor of 4. If the step size is reduced by a factor ofthe error is reduced by a factor ofThe number of calculations increases by a factor of k, and there is a balance to be struck between accuracy and the number of calculations.