Euler's Method

Numerical Analysis · Axiom Academy

The simplest numerical ODE solver Euler's method is beautifully simple: use the tangent line to approximate the curve. From any point, follow the slope for a small step. Here, h is the step size (how far we step in t), and f(tₙ, yₙ) is the slope at our current point—exactly what the ODE tells us! Euler's method comes directly from the first-order Taylor expansion: The exact solution satisfies y'(tₙ) = f(tₙ, y(tₙ)). Dropping higher-order terms gives us Euler's method. Let's solve the IVP: y' = y, y(0) = 1 (solution: y = eᵗ) using h = 0.5 for t ∈ [0, 2]. The exact value at t = 2 is e² ≈ 7.389, but Euler gives 5.0625—an error of about 31%! We need smaller h or better methods. Low accuracy: First-order method (error ~ h) Stability issues: Can diverge for stiff equations Error accumulation: Small errors compound over many steps Despite its limitations, Euler's method is fundamental because: It's the conceptual foundation for all ODE solvers It's perfect for teaching and prototyping It can work well for smooth problems with small enough h More sophisticated methods (RK4) are built on the same principle You've learned Euler's method—the simplest way to numerically solve ODEs!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.