Euler's Method

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