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Sachin Kaushal, Lovely Professional University Unit 14: Classification of Partial Differential Equations

Unit 14: Classification of Partial Differential Equations Notes

CONTENTS
Objectives
Introduction
14.1 Types of Differential Equations

14.2 Derivation of Partial Differential Equations
14.3 Various Classes of Partial Differential Equations
14.4 Summary
14.5 Keyword
14.6 Review Questions
14.7 Further Readings

Objectives

After studying this unit, you should be able to:

 Know before hand the type of the equation to be solved.
 Know that there are various methods based on the structure of the partial differential
equations.
 See that the partial differential equations of the first order are generally solved by methods
to get either complete solution or general solution.
 See that in the case of second order partial differential equations there are three types of
equations, i.e. hyperbolic type, parabolic type or elliptic type.
 Deal with the methods of dealing with various partial differential equations.

Introduction

The classification of the partial differential equations is quite different than those of ordinary
differential equations.
Some of the most important partial differential equations fall into one of the three categories
i.e., the hyperbolic type, the parabolic type or elliptic type.

14.1 Types of Differential Equations

In dealing with any differential equation involving a number of variables, we first of all classify
the variables into two categories. A variable may be such that it depends upon a number of other
variables. Such a variable is called dependent variable and the other variables on which it is
dependent are termed as independent variables.
In the case of ordinary differential equations we have to deal with one dependent and one
dy
independent variable. So the derivative of dependent variable is denoted as , where y is a
dx
dependent variable and x is an independent variable. So the differential equation may be of the
form

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