An Introduction to Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In simple terms, it describes how a certain quantity changes over time or space. Differential equations are widely used in fields such as physics, engineering, biology, and economics to model various dynamic systems.

First-Order Differential Equations

A first-order differential equation involves the first derivative of the unknown function but no higher-order derivatives. The general form is:

dy/dx = f(x, y)

Where y is the unknown function of x, and f(x, y) is some given function.

Second-Order Differential Equations

A second-order differential equation involves the second derivative of the unknown function. The general form is:

d²y/dx² = f(x, y, dy/dx)

These types of equations often arise in physics, particularly in systems involving motion, such as the motion of a spring or an oscillating pendulum.

Applications of Differential Equations

Differential equations are fundamental in modeling real-world phenomena such as:

• Physics: Modeling motion (Newton’s laws), heat flow, and wave propagation.
• Biology: Modeling population dynamics and the spread of diseases.
• Engineering: Describing electrical circuits and mechanical vibrations.
• Economics: Modeling growth rates, inflation, and market dynamics.