Page 349 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Sachin Kaushal, Lovely Professional University

Notes Unit 21: Classifications of Second Order
Partial Differential Equations

CONTENTS
Objectives

Introduction
21.1 Classification of Linear, second order Partial Differential Equations in Two
Independent Variables
21.2 Canonical form
21.3 Classification of Second Order Partial Differential Equations

21.4 Summary
21.5 Keyword

21.6 Review Questions
21.7 Further Readings

Objectives

After studying this unit, you should be able to:

 Observe that the partial differential equations of the second order can be of linear type or
non-linear type.
 Understand that linear partial differential equations can be classified into three categories,
namely hyperbolic, parabolic and elliptic type.
 Know that we have equations having variable coefficients there are some cases where the
equations involve variable coefficients but they can be transformed into equations with
constant coefficients.
Introduction

Classification of the partial differential equations help us in solving them in a systematic way. It
is advisable to understand the type of the partial differential equation before trying to solve it.
The methods of solving various classes of differential equations are also different.

21.1 Classification of Linear, Second Order Partial Differential
Equations in two Independent Variables

Consider a second order linear partial differential equation in two independent variables x and
y which can be written as
2  2  2   
x
y
y
x
y
x
y
y
x
x
a ( , ) 2 2 ( , ) c ( , ) 2 d 1 ( , ) d 2 ( , ) d 3 ( , ) f ( , ) ...(1)
b
y
x
y
x

x   x y y  x  x 
342 LOVELY PROFESSIONAL UNIVERSITY