Euler's Method

A numerical technique for approximating solutions to differential equations by stepping along the slope field in small increments.

Euler's method follows the current slope in straight steps.

Definition

Euler's Method is a numerical technique for approximating the solution to a differential equation. Given an initial value problem $\frac{dy}{dx} = f(x, y)$ with $y(x_0) = y_0$ , we step forward in small increments of size $h$ :

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$

At each step, we use the current slope $f(x_n, y_n)$ to predict where the curve goes next. It is the simplest numerical ODE solver and the foundation for understanding more advanced methods.

Key terms:

$h$ — the step size (smaller $h$ gives more accuracy)
$f(x_n, y_n)$ — the slope at the current point
$(x_n, y_n)$ — the current approximation point

One step of Euler's Method

Approximate $y(0.1)$ given $\frac{dy}{dx} = y$ , $y(0) = 1$ , using step size $h = 0.1$ .

$y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.1 \cdot (1) = 1.1$

The exact answer is $y(0.1) = e^{0.1} \approx 1.10517$ , so the error is about $0.005$ .

Try it

Use Euler's Method with $h = 0.5$ to approximate $y(1)$ given $\frac{dy}{dx} = x$ , $y(0) = 0$ .

Solution

Step 1: $x_0 = 0$ , $y_0 = 0$ . Slope $= f(0, 0) = 0$ . $y_1 = 0 + 0.5 \cdot 0 = 0$

Step 2: $x_1 = 0.5$ , $y_1 = 0$ . Slope $= f(0.5, 0) = 0.5$ . $y_2 = 0 + 0.5 \cdot 0.5 = 0.25$

So $y(1) \approx 0.25$ . The exact answer is $y = \frac{x^2}{2}$ , giving $y(1) = 0.5$ . Error $= 0.25$ — large because $h$ is large.

Related concepts

Needs first

Derivatives

Gradient Descent

View in full concept graph →