Differential Equation

Introduction

A Differential Equation is a mathematical equation that relates some function with its derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between them. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Types of Differential Equations

Differential equations can be broadly classified into two types: ordinary differential equations (ODEs) and partial differential equations (PDEs).

Ordinary Differential Equations

An Ordinary Differential Equation (ODE) contains functions of only one independent variable, and its derivatives. The term "ordinary" is used in contrast with the term "partial differential equation" which may be with respect to more than one independent variable.

Partial Differential Equations

A Partial Differential Equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

Linear Differential Equations

A differential equation is said to be linear if it is linear in the variables and their derivatives. A linear differential equation is of the form: dy/dx + p(x)y = q(x) where p(x) and q(x) are functions of x. The solution to this equation is given by the integrating factor method.

Homogeneous Differential Equations

A differential equation is said to be homogeneous if the degree of each term is the same. A homogeneous differential equation is of the form: dy/dx = f(y/x) or dy/dx = f(x/y) The solution to this equation is given by the substitution method.

Exact Differential Equations

An exact differential equation is of the form: M(x, y)dx + N(x, y)dy = 0 where M and N are functions of x and y. The solution to this equation is given by the method of exact differentials.

Bernoulli Differential Equations

A Bernoulli differential equation is of the form: dy/dx + p(x)y = q(x)y^n where p(x) and q(x) are functions of x and n is a real number. The solution to this equation is given by the substitution method.

Applications of Differential Equations

Differential equations have a wide range of applications in various fields. They are used to model a multitude of physical, biological, and social systems and to solve problems in engineering, physics, economics, and biology.

In Physics

Differential equations play a significant role in physics as they can describe physical phenomena such as motion, waves, heat flow, etc. For example, Newton's second law of motion, which states that the force acting on a body is equal to the mass of the body times its acceleration, can be written as a differential equation.

In Engineering

In engineering, differential equations are used to model systems such as electrical circuits, fluid flow, etc. For example, the behavior of an electrical circuit can be described using differential equations.

In Economics

In economics, differential equations are used to model economic behavior. For example, they can be used to model the change in investment or consumption over time.

In Biology

In biology, differential equations are used to model biological processes such as population growth, spread of diseases, etc. For example, the spread of a disease in a population can be modeled using a differential equation.