Difference Between Linear And Non-Linear Differential Equations

Introduction

Differential equations have played an essential role in the story of mathematics since the discovery of calculus in the 18th century by mathematicians such as Newton and Leibnitz. Because of their wide range of applications, differential equations are extremely important in mathematics. 

A differential equation is an equation that contains at least one differential coefficient or derivative of an unknown variable. There are two types of differential equations: linear and non-linear. 

This article aims to describe what a linear differential equation is, what a nonlinear differential equation is, and how linear and nonlinear differential equations are different.

Linear Differential Equations

A differential equation without nonlinear terms of the unknown function y and its derivatives is known as a linear differential equation.

For example: f: X→Y and f(x) = y.

It specifies that y cannot have higher index terms such as y², y³, and derivative multiples such as:

It also cannot contain non-linear terms such as . It takes the form, 

where y and g are functions of x. The equation is an order n differential equation, with n being the index of the highest order derivative. 

The differential operator in a linear differential equation is a linear operator, and the solutions are in the form of a vector space. A linear combination of the solutions is also a solution to the differential equation due to the linear nature of the solution set. If y1 and y2 are differential equation solutions, then is a solution as well.

The equation's linearity is simply one of the categorization parameters; it can also be classified as homogeneous or non-homogeneous and ordinary or partial differential equations. The equation is a linear homogeneous differential equation if the function is g=0. The equation is a linear partial differential equation if f is a function of two or more independent variables.

The differential equation's solution method is determined by the type and coefficients of the differential equation. When the coefficients are constant, the case is the simplest. Newton's second law of motion and its varied applications are a classic example of this case.

Non-Linear Differential Equations

Equations that contain nonlinear terms are known as non-linear differential equations. 

All of the equations listed above are nonlinear differential equations. Because nonlinear differential equations are difficult to solve, a thorough examination is necessary to arrive at a valid solution. When it comes to partial differential equations, the vast majority of them have no general solution. As a result, each equation must be dealt with separately.

Nonlinear partial differential equations include the Navier-Stokes equation and Euler's equation in fluid dynamics, as well as Einstein's field equations in general relativity. When the Lagrange equation is applied to a variable system, it might result in a system of nonlinear partial differential equations.

Now we will learn the major differences between linear and nonlinear differential equations.