Euler's Method

Euler's Method

Differential equations describe how quantities change over time. A first-order ODE has the form:

$\frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0$

We cannot always solve this analytically. Euler's method is the simplest numerical approach: approximate the solution by taking small steps along the tangent line.

The Formula

Given the current state $(t, y)$, take a small step of size $h$:

$y_{\text{next}} = y + h \cdot f(t, y)$

The idea: $f(t, y)$ is the slope (derivative) at the current point. Moving $h$ forward in time, we step $h \times \text{slope}$ in the y-direction.

Example

For $\frac{dy}{dt} = y$ (exponential growth) with $y(0) = 1$ and $h = 0.1$:

$y_{\text{next}} = 1 + 0.1 \times 1 = 1.1$

The exact answer at $t = 0.1$ is $e^{0.1} \approx 1.10517$. Euler's method gives $1.1$ — a reasonable approximation for a single step.

Your Task

Implement euler_step(f, t, y, h) that returns the next y value using one Euler step.