DIFFERENTIAL EQUATIONS

Differential Equations



Differential Equations

Application of Nonlinear Differential Equations

Introduction

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations of basic important in engineering mathematics, because physical laws and relations appear mathematically in the form of a differential equation, and in other disciplines. In additional, Difference equations are mathematically studied from several different perspectives, mostly concerned with their solutions - the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated us- ing computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of ac- curacy. The theory of differential equations is quite developed and the methods used to study them vary signi.cantly with two types of the equation which are Ordinary Differential Equation, and Partial Differential Equation.

Ordinary Differential Equation (ODE)

An equation which contains the derivatives of a yet to be determined function y(x) (a function of one variable) an ordinary differential equation ODE in y(x) of 1st order is defined to:

First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations.

An ODE in y(x) of second order is de…ned to:

In the same method, we will get ODE in y(x) of 3rd……. nth order is defines to:

3. Partial Differential Equation

A PDE is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is de…ned similarly to the case of ordinary differential equations, but further classi…cation into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial di¤er- ential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

3.1. Linear Difference Equation

A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coe¢ cients are functions of independent variables.

3.2. Non Linear Differennce Equation

The non-linear differential equations is a technique derived from a much broader concept which are differential equations, but even when a topic is derived from the other, there are differences between the time of application, for example, differential equations linear …rst order may have unique solutions, while the nonlinear equations do not have. Nonlinear difference equations aims to explain simply and teaching the theoretical surrounding, the development, resolution and implementation of Difference Equations. Nonlinear systems represent sys- tems whose behavior is not expressible as the sum of the behaviors of its de- scriptors.

More formally, a physical system, mathematical or otherwise is ...