Euler Method

The Euler method is the simplest numerical technique for solving first-order ordinary differential equations (ODEs) with an initial condition. It approximates the solution of dy/dx = f(x, y), y(x₀) = y₀ by advancing the solution step-by-step using the slope at the current point. The method is derived from the Taylor series expansion: y(x + h) = y(x) + h·y'(x) + (h²/2)·y''(x) + ... . Truncating after the first-order term gives y(x + h) ≈ y(x) + h·y'(x). Since y'(x) = f(x, y), the update formula becomes y_{n+1} = y_n + h·f(x_n, y_n), where h is the step size. The Euler method is explicit (the new value y_{n+1} is computed directly from known values) and first-order accurate. The local truncation error (error per step) is O(h²), while the global truncation error (accumulated error over the entire interval) is O(h). This means halving the step size approximately halves the global error, but also doubles the number of steps required. To apply the method: (1) Specify the ODE dy/dx = f(x, y) and initial condition y(x₀) = y₀. (2) Choose a step size h (smaller h gives better accuracy but more computation). (3) Iterate: x_{n+1} = x_n + h, y_{n+1} = y_n + h·f(x_n, y_n). (4) Continue until the desired x value is reached. The Euler method has limitations: (1) Low accuracy compared to higher-order methods (Runge-Kutta, multistep methods). (2) Stability constraints: for stiff ODEs or rapidly decaying solutions, h must be very small to avoid instability. (3) Error accumulates over many steps. Despite these limitations, Euler's method is useful for: introducing numerical ODE concepts, quick estimates, problems where low accuracy is acceptable, and as a building block for more sophisticated methods (e.g., improved Euler, Runge-Kutta). The improved Euler method (Heun's method, explicit trapezoidal) uses an average of slopes: y_{n+1} = y_n + (h/2)[f(x_n, y_n) + f(x_{n+1}, y_n + h·f(x_n, y_n))], which is second-order accurate (global error O(h²)).