# 19.3: Euler’s Method


Here’s a test to see if you’re as smart as Leonhard Euler. Let’s say you arrive at time ( $$t$$) and measure the current population ($$y$$) and the rate of change ($$r$$). What do you think the population will be after some period of time $$\Delta t$$ has elapsed?

If you said $$y + r \Delta t$$, congratulations! You just invented Euler’s method.

This estimate is based on the assumption that $$r$$ is constant, but in general it’s not, so we only expect the estimate to be good if $$r$$ changes slowly and $$\Delta t$$ is small.

What if we want to make a prediction when $$\Delta t$$ is large? One option is to break $$\Delta t$$ into smaller pieces, called time steps. Then we can use the following equations to get from one time step to the next: \begin{aligned} T_{i+1} &=& T_i + dt \\ Y_{i+1} &=& Y_i + \frac{df}{dt}(t) \, dt\end{aligned}

Here, $$T_i$$ is a sequence of times where we estimate the value of $$y$$, and $$Y_i$$ is the sequence of estimates. For each index $$i$$, $$Y_i$$ is an estimate of $$y(T_i)$$.

If the rate doesn’t change too fast and the time step isn’t too big, Euler’s method is accurate enough for most purposes.

This page titled 19.3: Euler’s Method is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.