Introduction to Differential Equations

A differential equation is an equation that contains an unknown function and one or more of its derivatives. The order of a differential equation is given by the highest order derivative present in the equation.

When the equation involves only the first derivative, it is called first order. If it involves the second derivative, it is called second order, and so on.

For example, a spring-mass system can be modeled as a simple harmonic oscillator, which equates the net force on the mass and the restoring force in the spring, described by Hooke’s law for linear springs.

The spring force acts in the direction opposite to the displacement and is proportional to how far the mass moves from its equilibrium position.

Newton’s second law links the net force on the mass to its acceleration. Combining these two relationships gives an equation in which mass times acceleration equals a negative constant times displacement.

By rewriting acceleration as the second derivative of displacement over time, a second-order differential equation is obtained.

This second derivative links the motion of the system to a differential equation.