Differential Equation and its Application in Real Life (Prakash Bayalkoti)

 Differential Equation

and

its Application in Real Life

Prakash Bayalkoti 

Department of Mathematics

An equation that connects one or more unknown functions and their derivatives is known as a differential equation in mathematics. In practical applications, the differential equation 

establishes a connection between the functions, which often reflect physical values, and the 

derivatives, which represent their rates of change. Differential equations are important in 

many fields, including as engineering, physics, economics, and biology, since these kinds of 

relationships are frequently found in mathematical models and scientific principles. 

The primary focus of studying differential equations is on their solutions, or the collection of 

functions that satisfy each equation, as well as the characteristics of those solutions. Although many aspects of solutions to a given differential equation can be ascertained without precisely  computing them, only the most basic differential equations can be solved using explicit formulas.

Introduction: 

Differential equations came into existence with the invention of calculus by Isaac

Newton and Gottfried Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Newton listed three kinds of differential equations:

In all these cases, y is an unknown function of x (or of x1 and x2), and f is a given function. He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

Types of DE: Different types of differential equations can be distinguished. These 

classes of differential equations not only describe the characteristics of the equation but 

also aid in guiding the selection of a solution strategy. Whether the equation is ordinary 

or partial, linear or non-linear, homogeneous or heterogeneous, are examples of 

frequently used distinctions.

• An equation with an unknown function of one real or complex variable x, its derivatives, and certain provided functions of x is called an ordinary differential equation (ODE).

• A partial differential equation (PDE) is a differential equation that contains 

unknown multivariable functions and their partial derivatives.

• If a system of differential equations is not a system of linear equations, it is referred to as nonlinear. Nonlinear differential equation problems come in a wide variety, and approaches to solving or analysing them vary depending on the situation. The Lotka- Volterra equations in biology and the Navier-Stokes equations in fluid dynamics are two instances of nonlinear differential equations.

Applications: 

Differential equations are a broad topic in engineering, physics, and pure and applied 

mathematics. The characteristics of different kinds of differential equations are the focus of all these fields. Whereas practical mathematics stresses the rigorous explanation of the 

techniques for approximating solutions, pure mathematics concentrates on the existence and uniqueness of solutions. In order to model almost any physical, technical, or biological 

process—from the motion of the stars to the design of bridges to the interactions between 

neurons—differential equations are essential. Differential equations, like those used to solvereal-world issues, do not always have closed-form solutions, or be directly solvable. 

Alternatively, numerical techniques can be used to approximate solutions. 

Differential equations can be used to formulate many basic rules of physics and chemistry. 

Differential equations are used to model the behaviour of complex systems in both biology 

and economics. Differential equations' mathematical theory initially evolved alongside the 

fields from whence the equations came and where the findings were used. Nonetheless, 

various issues that may come from very different scientific domains might result in differential 

equations that are exactly the same. When this occurs, the mathematical theory underlying 

the equations can be seen as a commonality among various events. Take, for instance, how 

sound and light travel through the atmosphere and how waves move across a pond's surface. 

We can think of light and sound as kinds of waves, similar to the well-known waves in water, 

because they can all be represented by the same second-order partial differential equation, 

known as the wave equation. Joseph Fourier established the theory of conduction of heat, 

which is governed by the heat equation, another second-order partial differential equation. 

The Black-Scholes equation in finance, for example, is connected to the heat equation. It 

turns out that many diffusion processes, despite their apparent differences, are characterised 

by the same equation.