Euler’s Method

Euler’s Method helps you approximate solutions to differential equations. You’ll always be given an equation and an initial condition, and you’ll use this information to carry out the Euler’s approximation in multiple steps. The better approximation you want, the more steps you have to take. The only formula you’ll need to remember is

In this formula, y1 represents the first y-value you’ll need to calculate. y0 and t0 are the values you’re given in the initial condition, and (delta)t is the distance between values of t.

Note that the values of t and y given in the initial condition are denoted by t0 and y0. This does not necessarily mean that t=0 or that y=0. This only means that these values of t and y were given in the original problem.

Each subsequent approximation of y is denoted by y1, y2, y3, … yn and each value of t for which y is estimated is denoted by t1, t2, t3, … tn. The “step-size” or the distance between two successive values of t at which y is approximated is often denoted by h or (delta)t.

Let’s take a look at the example on the left.

We’ll also look at one more example of a problem worded slightly differently.

One thing you MUST remember whenever you’re using Euler’s Method, is to keep ALL of your decimal places until you get to the last value of , and you’ve found your final answer. Remember that Euler’s Method is about approximation, so if you start rounding off decimal places prior to the end of the problem, you’re approximation will get less and less accurate as you go. Therefore, if you’re calculator gives you twelve decimal places, write down every single one until you get your final answer. THEN, and ONLY then, can you round your answer.