Euler's method, proposed by Leonhard Euler in 1768, is the oldest and conceptually simplest algorithm for numerically solving ordinary differential equations. Given the derivative function f(t, y) and a current state (t_n, y_n), it estimates the next state as y_{n+1} = y_n + h·f(t_n, y_n), where h is the timestep. Each step follows the tangent line to the solution curve for a distance h before recomputing the slope. Despite its simplicity, the method is first-order accurate — global error grows as O(h) — and all higher-order methods like Runge-Kutta can be understood as improvements upon this basic idea.
The key limitation of Euler's method is error accumulation. For a simple test ODE dy/dt = −k·y (exponential decay), the exact solution is y₀·e^(−kt), but each Euler step introduces a local truncation error proportional to h²·y''. This error accumulates step by step, so after N steps the global error scales as N·h² = T·h, where T is total simulation time — confirming first-order convergence. Halving the step size halves the final error but doubles the computation, while RK4 achieves fourth-order convergence with only four times the work per step.
This interactive visualization solves a chosen ODE simultaneously with Euler's method and the exact analytical solution, letting you watch the numerical error accumulate in real time. Adjusting the step size h reveals the tradeoff directly: large h computes fast but the Euler trajectory visibly drifts from the true solution, while small h tracks the exact curve closely at the cost of more steps. This makes the tool a clear demonstration of why step-size selection is one of the fundamental practical decisions in numerical ODE solving.