Euler's Method

Definition

Euler’s Method is a fundamental numerical procedure for approximating the solution to a first-order initial value problem (IVP). It works by starting at a known point $(x_0, y_0)$ and taking small, discrete steps along the tangent line defined by the differential equation $\frac{dy}{dx} = f(x, y)$.

• How to read: “The derivative d y divided by d x equals f of x comma y.”
• Meaning: Slope at each point comes from the DE—Euler follows that local slope in steps of size $\Delta x$.

Why It Matters

Euler’s method is the fundamental “bridge” between continuous differential equations and the discrete steps of a digital computer. It serves as the primary algebraic substitute for real-time physics engines and biological simulations, allowing us to trace the trajectory of falling objects or the spread of disease frame-by-frame.

Core Concepts

• Iteration Formula: The next value is calculated as $y_{n+1} = y_n + f(x_n, y_n) \Delta x$, where $\Delta x$ is a fixed step size.
  • How to read: “The next value y n plus one equals y n plus f evaluated at x n and y n, multiplied by delta x.”
  • Meaning / when to use: Euler step—follow the tangent slope for one small step; repeat to trace an approximate solution curve to a DE.
• Linear Approximation: The method assumes the slope is constant over the interval $\Delta x$, which is the source of its approximation error.
• Step Size Sensitivity: Smaller step sizes ($\Delta x$) generally lead to more accurate approximations but require more computational steps.

Connected Concepts

• The Closed Interval Method
• Lagrange Multipliers Method
• Method of Variation of Parameters