Introduction to Differential Equations

A differential equation is a mathematical expression that establishes a relationship between a function and its derivatives. These equations are fundamental in modeling dynamic systems across various fields of science and engineering. The order of a differential equation is defined by the highest order derivative present in the equation. A first-order differential equation includes only the first derivative, while a second-order differential equation includes up to the second derivative of the unknown function.

Second-Order Differential Equations in Physical Systems

A classic example of a second-order differential equation arises from modeling the motion of a spring–mass system. In this idealized system, a mass is attached to a spring that obeys Hooke’s Law and is free from any damping or external forces. When the mass is displaced from its equilibrium position, the spring exerts a restoring force that is linearly proportional to the displacement and acts in the opposite direction. Mathematically, this restoring force is given by

where k is the spring constant and x is the displacement from equilibrium.

According to Newton’s second law, the net force acting on an object is equal to the product of its mass and acceleration. Substituting the spring force into Newton’s law yields

When acceleration is expressed as the second derivative of displacement with respect to time.

Rearranging terms leads to the standard form of the second-order linear differential equation:

This equation describes simple harmonic motion and serves as a foundational model in the study of mechanical vibrations and oscillatory systems.