An equation relating one or more functions and their derivatives can be termed as a differential equation. In general, the function represents physical quantities whereas the derivatives define the rate of change, the differential equation explains the relation between the two. The concept of differential equations plays an important role in the field of physics, economics and engineering. The topic of differential equations saw its existence after the invention of calculus by the great scientists’ Newton and Leibniz.
The general form of an ordinary differential equation is as follows.
y’ + P (x) y = Q (x) yn
The various types of differential equations are explained below.
1] Ordinary differential equations
It is an equation that consists of 1 unknown function of a variable x, its derivatives and a few functions of x. y denotes the unknown function that depends on another variable x. Here x is the independent variable.
2] Linear differential equations
The differential equations that are linear in the derivatives and the unknown functions can be called linear differential equations.
3] Partial differential equations
The differential equation containing the unknown multivariable functions and its partial derivatives are partial differential equations. They are used to formulate problems with functions consisting of many variables and are solved in a closed-form.
Partial differential equations find its applications in phenomena of nature namely heat, sound, elasticity or electrostatics.
4] Nonlinear differential equations
A non-linear differential equation in the function that is unknown and its derivatives are nonlinear differential equations. Some of the methods used to solve nonlinear differential equations include the ones that are typically dependent on the equation possessing particular symmetries. They demonstrate complicated behaviour on time intervals that are extended.
5] Equation order
The differential equations that can be categorised by their order, finding the term with the highest derivatives.
6] Bernoulli differential equations
The differential equations of the form y’ + p (t) y = yn.
7] Algebraic differential equations
A differential equation that includes differential and algebraic terms in implicit form is an algebraic differential equation.
8] Integro differential equations
A differential equation consisting of a differential equation and an integral equation.
9] Stochastic differential equations
An equation in which the quantity that is unknown is a stochastic process and also involves a few known stochastic processes.
Applications of differential equations
1] The concept of differential equations finds its vast applications in the field of physics, pure and applied maths.
2] Pure mathematics deals with the existence and uniqueness of solutions, whereas applied mathematics talks about the rigorous justification of the methods for approximating solutions.
3] In the modelling of each physical, technical or a biological process, differential equations are used.
4] Most of the principal laws of chemistry and physics are formulated using the differential equations.
5] To monitor the behaviour of complex systems, differential equations are used.
6] The conduction of heat which was proposed by Joseph Fourier is based on the second-order differential equation – that is the heat equation.
7] The wave equation can be described by the second-order partial differential equations.
There are wide applications of differential equations. A few of them are listed above for reference. To obtain more information on differential equations, quadratic inequalities, integration, differentiation, calculus and algebra please visit BYJU’S website.