Lecture 7 - First Order Partial Differential Equations | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
Subject: Math  
Lesson: First Order Partial Differential Equations 
Course Developer: Manoj Kumar 
College/Department: Hans Raj College (D.U.) 
 
 
 
 
 
  
Page 2


First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
Subject: Math  
Lesson: First Order Partial Differential Equations 
Course Developer: Manoj Kumar 
College/Department: Hans Raj College (D.U.) 
 
 
 
 
 
  
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
Table of Contents: 
 Chapter : First Order Partial Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Classification of partial differential equations of order one 
? 3.1: Linear partial differential equations 
? 3.2: Semi linear partial differential equations 
? 3.3: Quasi linear partial differential equations 
? 3.4: Non linear partial differential equations 
? 4: Construction of partial differential equations of first order 
? 4.1: Elimination of arbitrary constants 
o Exercises 
? 4.2: By elimination of arbitrary functions 
o Exercises 
? 5: Classification of Integrals or Solutions 
? 6: Geometrical interpretation of partial differential equation of 
first order 
? 7: Method of Characteristics 
? 8: The Cauchy problem for partial differential equation of first 
order 
o Exercises 
? 9: Canonical Forms of Linear Equations of First Order 
o Exercise 
? 10: Method of Separation of Variables 
o Exercises 
? Summary 
? Glossary 
? References/ Further Reading 
Page 3


First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
Subject: Math  
Lesson: First Order Partial Differential Equations 
Course Developer: Manoj Kumar 
College/Department: Hans Raj College (D.U.) 
 
 
 
 
 
  
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
Table of Contents: 
 Chapter : First Order Partial Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Classification of partial differential equations of order one 
? 3.1: Linear partial differential equations 
? 3.2: Semi linear partial differential equations 
? 3.3: Quasi linear partial differential equations 
? 3.4: Non linear partial differential equations 
? 4: Construction of partial differential equations of first order 
? 4.1: Elimination of arbitrary constants 
o Exercises 
? 4.2: By elimination of arbitrary functions 
o Exercises 
? 5: Classification of Integrals or Solutions 
? 6: Geometrical interpretation of partial differential equation of 
first order 
? 7: Method of Characteristics 
? 8: The Cauchy problem for partial differential equation of first 
order 
o Exercises 
? 9: Canonical Forms of Linear Equations of First Order 
o Exercise 
? 10: Method of Separation of Variables 
o Exercises 
? Summary 
? Glossary 
? References/ Further Reading 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
 
1. Learning Outcomes: 
After reading this chapter, learner will be able to understand  
? How to construct a  first order PDE 
? How to solve a first order PDE using the method of characteristics 
? How to transform a PDE of first order in canonical form. 
? How to solve PDE of first order using the method separation of 
variables.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. Introduction: 
Page 4


First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
Subject: Math  
Lesson: First Order Partial Differential Equations 
Course Developer: Manoj Kumar 
College/Department: Hans Raj College (D.U.) 
 
 
 
 
 
  
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
Table of Contents: 
 Chapter : First Order Partial Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Classification of partial differential equations of order one 
? 3.1: Linear partial differential equations 
? 3.2: Semi linear partial differential equations 
? 3.3: Quasi linear partial differential equations 
? 3.4: Non linear partial differential equations 
? 4: Construction of partial differential equations of first order 
? 4.1: Elimination of arbitrary constants 
o Exercises 
? 4.2: By elimination of arbitrary functions 
o Exercises 
? 5: Classification of Integrals or Solutions 
? 6: Geometrical interpretation of partial differential equation of 
first order 
? 7: Method of Characteristics 
? 8: The Cauchy problem for partial differential equation of first 
order 
o Exercises 
? 9: Canonical Forms of Linear Equations of First Order 
o Exercise 
? 10: Method of Separation of Variables 
o Exercises 
? Summary 
? Glossary 
? References/ Further Reading 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
 
1. Learning Outcomes: 
After reading this chapter, learner will be able to understand  
? How to construct a  first order PDE 
? How to solve a first order PDE using the method of characteristics 
? How to transform a PDE of first order in canonical form. 
? How to solve PDE of first order using the method separation of 
variables.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. Introduction: 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
An equation is called a partial differential equation if it contains one or more 
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and 
Euler started the study of partial differential equations (PDE's) as the 
principal mode of analytical study of models in the physical science. In the 
middle of the 19
th
 century, PDE's also became a necessary tool in other 
branches of mathematics. 
 
3. Classification of partial differential equations of order 
one: 
The most general form of the partial differential equations of order one in 
two independent variables   and   is given by  
 (       
 
  
 
)      
Or 
 (         )                           (1) 
Where   is a function of two independent variables  ,   and dependent 
variable   and its partial derivatives  
 
  
  
  
     
 
  
  
  
  . 
 
Similarly, if there are three independent variables         then the partial 
differential equations of order one can be written as 
  (       
 
  
 
  
 
)                                    (2) 
3.1. Linear partial differential equations: 
A partial differential equation  (         )   is said to be linear if the 
degree of the dependent variable   and all its partial derivatives i.e.       is 
one. 
i.e. if it is of the form 
  (   ) 
 
  (   ) 
 
  (   )   (   ) 
Where  (   )  (   )  (   ) and  (   ) are functions of        only. 
3.1.1. Examples of linear partial differential equations: 
(a)  
 
 
 
    
 
 (   )   
 
  
 
 
Page 5


First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
Subject: Math  
Lesson: First Order Partial Differential Equations 
Course Developer: Manoj Kumar 
College/Department: Hans Raj College (D.U.) 
 
 
 
 
 
  
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
Table of Contents: 
 Chapter : First Order Partial Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Classification of partial differential equations of order one 
? 3.1: Linear partial differential equations 
? 3.2: Semi linear partial differential equations 
? 3.3: Quasi linear partial differential equations 
? 3.4: Non linear partial differential equations 
? 4: Construction of partial differential equations of first order 
? 4.1: Elimination of arbitrary constants 
o Exercises 
? 4.2: By elimination of arbitrary functions 
o Exercises 
? 5: Classification of Integrals or Solutions 
? 6: Geometrical interpretation of partial differential equation of 
first order 
? 7: Method of Characteristics 
? 8: The Cauchy problem for partial differential equation of first 
order 
o Exercises 
? 9: Canonical Forms of Linear Equations of First Order 
o Exercise 
? 10: Method of Separation of Variables 
o Exercises 
? Summary 
? Glossary 
? References/ Further Reading 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
 
1. Learning Outcomes: 
After reading this chapter, learner will be able to understand  
? How to construct a  first order PDE 
? How to solve a first order PDE using the method of characteristics 
? How to transform a PDE of first order in canonical form. 
? How to solve PDE of first order using the method separation of 
variables.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. Introduction: 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
An equation is called a partial differential equation if it contains one or more 
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and 
Euler started the study of partial differential equations (PDE's) as the 
principal mode of analytical study of models in the physical science. In the 
middle of the 19
th
 century, PDE's also became a necessary tool in other 
branches of mathematics. 
 
3. Classification of partial differential equations of order 
one: 
The most general form of the partial differential equations of order one in 
two independent variables   and   is given by  
 (       
 
  
 
)      
Or 
 (         )                           (1) 
Where   is a function of two independent variables  ,   and dependent 
variable   and its partial derivatives  
 
  
  
  
     
 
  
  
  
  . 
 
Similarly, if there are three independent variables         then the partial 
differential equations of order one can be written as 
  (       
 
  
 
  
 
)                                    (2) 
3.1. Linear partial differential equations: 
A partial differential equation  (         )   is said to be linear if the 
degree of the dependent variable   and all its partial derivatives i.e.       is 
one. 
i.e. if it is of the form 
  (   ) 
 
  (   ) 
 
  (   )   (   ) 
Where  (   )  (   )  (   ) and  (   ) are functions of        only. 
3.1.1. Examples of linear partial differential equations: 
(a)  
 
 
 
    
 
 (   )   
 
  
 
 
First Order Partial Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      pg. 5 
 
(b)   
 
   
 
       
(c)  ( 
 
  
 
) 
 
 (     
 
) 
 
 (   )     (   ) 
3.2. Semi linear partial differential equations: 
A partial differential equation of order one  (         )   is said to be semi 
linear if it is of the form 
  (   ) 
 
  (   ) 
 
  (     ) 
Where  and   are the functions of        only. 
3.2.1. Examples of semi linear partial differential equations: 
(a)   
 
   
 
   
 
 
(b) (   )
 
 
 
 (   )
 
 
 
 (   ) 
 
 
(c)   
 
 
 
  
 
 
 
 (   )  
3.3. Quasi linear partial differential equations: 
A partial differential equation  (         )   is said to be quasi linear if the 
degree of the all partial derivatives i.e.       is one. i.e. if it is of the form 
 (     ) 
 
  (     ) 
 
  (     ) 
Where      and   are the functions of        . 
3.3.1. Examples of quasi linear partial differential equations: 
(a) ( 
 
  
 
) 
 
    
 
    
(b) ( 
 
  
 
) 
 
    
 
  
 
   
 
 
(c)  ( 
 
   ) 
 
 ( 
 
   ) 
 
  
 
    
3.4. Non linear partial differential equations: 
A partial differential equation  (         )   is said to be non linear if the 
degree of the partial derivatives i.e.       is not one. 
3.4.1. Examples of non-linear partial differential equations:  
(a)  
 
 
 
   
(b)  
 
(   
 
 
  
 
 
)  
 
 
(c)