Source Code for Solving Second-Order Differential Equations Using Euler's Method

Euler's method serves as a fundamental numerical technique frequently employed to obtain approximate solutions for differential equation systems. For second-order differential equations, the typical approach involves transforming them into a system of first-order equations through variable substitution before applying Euler's iterative step-by-step solving method.

Taking the classic spring-mass oscillator system as an example, its equation of motion can be expressed as a second-order differential equation. By introducing velocity variables, the original equation can be decomposed into a system of two first-order differential equations. The core concept of Euler's method utilizes current state variables to estimate the next state through linear approximation.

In practical implementation, the first step requires defining initial conditions including initial displacement and initial velocity. Subsequently, using a specified time step, the displacement and velocity values are sequentially updated at each time point. The displacement update depends on the current velocity, while the velocity update is determined by the current acceleration - which itself relates to the current displacement and velocity through the equation of motion. The implementation typically involves initializing arrays to store time, position, and velocity data, followed by a loop that calculates derivatives at each step and updates the state variables using the formula: y_new = y_old + h * f(t, y_old), where h represents the time step.

The primary advantage of this method lies in its straightforward implementation and computational efficiency, making it ideal as an introductory example for understanding numerical solution techniques. However, since Euler's method employs forward difference approximation, its accuracy and stability are significantly influenced by step size selection. Oversized steps may cause solution divergence, while excessively small steps increase computational costs. Therefore, practical applications often require balancing precision and efficiency when choosing appropriate step size parameters. Common implementations include error checking mechanisms and adaptive step size controls to improve reliability.