Lecture 3 - Solutions of Linear Homogeneous Differential equation | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
DC-1  
Sem-2 
Course: Diffrential Equation-1 
 
Lesson: Solutions of Homogeneous Linear 
Differential Equations 
Lesson Developer: Brijendra Yadav and Chaman 
Singh 
Department/College: Assistant Professor, 
Department of Mathematics, Acharya Narendra Dev 
College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Page 2


Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
DC-1  
Sem-2 
Course: Diffrential Equation-1 
 
Lesson: Solutions of Homogeneous Linear 
Differential Equations 
Lesson Developer: Brijendra Yadav and Chaman 
Singh 
Department/College: Assistant Professor, 
Department of Mathematics, Acharya Narendra Dev 
College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Page 3


Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
DC-1  
Sem-2 
Course: Diffrential Equation-1 
 
Lesson: Solutions of Homogeneous Linear 
Differential Equations 
Lesson Developer: Brijendra Yadav and Chaman 
Singh 
Department/College: Assistant Professor, 
Department of Mathematics, Acharya Narendra Dev 
College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Solutions of Homogeneous Linear Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Linear Differential Equations 
o 3.1: Second Order Linear Differential Equation  
? 4: Homogeneous and Non-homogeneous Linear Differential 
Equation 
o 4.1: Homogeneous and Non-homogeneous Linear 
Differential Equation with Constant Coefficients 
? 5: Principle of Superposition for Homogeneous Equations 
? 6: Existence and Uniqueness Theorem 
? 7: Linearly Independent or Linearly Dependent Functions 
o 7.1: Wronskian 
? 8: Solutions of Homogeneous Linear Differential Equations with 
Constant Coefficients 
o 8.1: Case (I): Roots of the auxiliary equation are real and 
distinct 
o 8.2: Case (II): Roots of the auxiliary equation are real but 
some roots are equal 
o 8.3: Case (III): Roots of the auxiliary equation are 
complex 
? 9: Euler Equation 
? Exercises 
? Summary 
? Reference 
 
 
 
 
 
 
 
Page 4


Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
DC-1  
Sem-2 
Course: Diffrential Equation-1 
 
Lesson: Solutions of Homogeneous Linear 
Differential Equations 
Lesson Developer: Brijendra Yadav and Chaman 
Singh 
Department/College: Assistant Professor, 
Department of Mathematics, Acharya Narendra Dev 
College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Solutions of Homogeneous Linear Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Linear Differential Equations 
o 3.1: Second Order Linear Differential Equation  
? 4: Homogeneous and Non-homogeneous Linear Differential 
Equation 
o 4.1: Homogeneous and Non-homogeneous Linear 
Differential Equation with Constant Coefficients 
? 5: Principle of Superposition for Homogeneous Equations 
? 6: Existence and Uniqueness Theorem 
? 7: Linearly Independent or Linearly Dependent Functions 
o 7.1: Wronskian 
? 8: Solutions of Homogeneous Linear Differential Equations with 
Constant Coefficients 
o 8.1: Case (I): Roots of the auxiliary equation are real and 
distinct 
o 8.2: Case (II): Roots of the auxiliary equation are real but 
some roots are equal 
o 8.3: Case (III): Roots of the auxiliary equation are 
complex 
? 9: Euler Equation 
? Exercises 
? Summary 
? Reference 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 4 
 
1. Learning Outcomes: 
After reading this lesson reader will be able to understand the following 
? Linear differential equation 
? Second order linear differential equation 
? Homogeneous and Non-homogeneous Linear Differential Equation 
? Homogeneous and Non-homogeneous Linear Differential Equation with 
Constant Coefficients 
? Principle of Superposition for Homogeneous Equations 
? Existence and Uniqueness Theorem 
? Linearly Independent or Linearly Dependent Functions 
? Wronskian 
? Solutions of Homogeneous Linear Differential Equations with Constant 
Coefficients 
? Euler Equation 
2. Introduction: 
Differential equations frequently appear as mathematical models of 
mechanical systems and electrical circuits. In this lesson, we will study about 
the homogeneous differential equations and their solutions. 
3. Linear Differential Equations: 
A differential equation of the form 
 ( , , ',y'', ..., ) ( )
n
F x y y y R x ?  
is called the linear differential equation provided that F is linear differential 
equation of order n in the dependent variable y and its derivatives 
',y'', ...,
n
yy . 
In other words, a differential equation of the form 
 
12
0 1 2 1 12
( ) ( ) ( ) ... ( ) ( ) ( )
n n n
nn n n n
d y d y d y dy
a x a x a x a x a x y R x
dx dx dx dx
??
? ??
? ? ? ? ? ?  (1) 
Where 
0 1 2 1
( ), ( ), ( ), ..., ( ), ( ) and ( )
nn
a x a x a x a x a x R x
?
are continuous functions of x 
only on some open interval I is called linear differential equation of order n. 
 
Page 5


Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
DC-1  
Sem-2 
Course: Diffrential Equation-1 
 
Lesson: Solutions of Homogeneous Linear 
Differential Equations 
Lesson Developer: Brijendra Yadav and Chaman 
Singh 
Department/College: Assistant Professor, 
Department of Mathematics, Acharya Narendra Dev 
College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Solutions of Homogeneous Linear Differential Equations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Linear Differential Equations 
o 3.1: Second Order Linear Differential Equation  
? 4: Homogeneous and Non-homogeneous Linear Differential 
Equation 
o 4.1: Homogeneous and Non-homogeneous Linear 
Differential Equation with Constant Coefficients 
? 5: Principle of Superposition for Homogeneous Equations 
? 6: Existence and Uniqueness Theorem 
? 7: Linearly Independent or Linearly Dependent Functions 
o 7.1: Wronskian 
? 8: Solutions of Homogeneous Linear Differential Equations with 
Constant Coefficients 
o 8.1: Case (I): Roots of the auxiliary equation are real and 
distinct 
o 8.2: Case (II): Roots of the auxiliary equation are real but 
some roots are equal 
o 8.3: Case (III): Roots of the auxiliary equation are 
complex 
? 9: Euler Equation 
? Exercises 
? Summary 
? Reference 
 
 
 
 
 
 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 4 
 
1. Learning Outcomes: 
After reading this lesson reader will be able to understand the following 
? Linear differential equation 
? Second order linear differential equation 
? Homogeneous and Non-homogeneous Linear Differential Equation 
? Homogeneous and Non-homogeneous Linear Differential Equation with 
Constant Coefficients 
? Principle of Superposition for Homogeneous Equations 
? Existence and Uniqueness Theorem 
? Linearly Independent or Linearly Dependent Functions 
? Wronskian 
? Solutions of Homogeneous Linear Differential Equations with Constant 
Coefficients 
? Euler Equation 
2. Introduction: 
Differential equations frequently appear as mathematical models of 
mechanical systems and electrical circuits. In this lesson, we will study about 
the homogeneous differential equations and their solutions. 
3. Linear Differential Equations: 
A differential equation of the form 
 ( , , ',y'', ..., ) ( )
n
F x y y y R x ?  
is called the linear differential equation provided that F is linear differential 
equation of order n in the dependent variable y and its derivatives 
',y'', ...,
n
yy . 
In other words, a differential equation of the form 
 
12
0 1 2 1 12
( ) ( ) ( ) ... ( ) ( ) ( )
n n n
nn n n n
d y d y d y dy
a x a x a x a x a x y R x
dx dx dx dx
??
? ??
? ? ? ? ? ?  (1) 
Where 
0 1 2 1
( ), ( ), ( ), ..., ( ), ( ) and ( )
nn
a x a x a x a x a x R x
?
are continuous functions of x 
only on some open interval I is called linear differential equation of order n. 
 
Solutions of Homogeneous Linear Differential Equations 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 5 
 
Value Addition: Note 
If the dependent variable y and all its derivatives ',y'', ...,
n
yy appear linearly 
in a differential equation then this equation is called linear differential 
equation. 
 
3.1. Second Order Linear Differential Equation: 
A differential equation of the form 
 
2
0 1 2 2
( ) ( ) ( ) ( )
d y dy
a x a x a x y R x
dx dx
? ? ? 
Where 
0 1 2
( ), ( ), ( ) and R(x) a x a x a x are continuous functions of x only on some 
open interval I is called second order linear differential equation. 
4. Homogeneous and Non-homogeneous Linear Differential 
Equation: 
If R(x) = 0, then the differential equation of the form 
 
12
0 1 2 1 12
( ) ( ) ( ) ... ( ) ( ) 0
n n n
nn n n n
d y d y d y dy
a x a x a x a x a x y
dx dx dx dx
??
? ??
? ? ? ? ? ? 
Where 
0 1 2 1
( ), ( ), ( ), ..., ( ) and ( )
nn
a x a x a x a x a x
?
are continuous functions of x only on 
some open interval I is called homogeneous linear differential equation of 
order n. 
If ( ) 0 Rx ? , then the differential equation of the form 
 
12
0 1 2 1 12
( ) ( ) ( ) ... ( ) ( ) ( )
n n n
nn n n n
d y d y d y dy
a x a x a x a x a x y R x
dx dx dx dx
??
? ??
? ? ? ? ? ? 
Where 
0 1 2 1
( ), ( ), ( ), ..., ( ), ( ) and ( )
nn
a x a x a x a x a x R x
?
are continuous functions of x 
only on some open interval I is called the non-homogeneous linear 
differential equation of order n. 
Value Addition: Second Order Homogeneous and Non-homogeneous 
Linear Differential Equation 
1. If R(x) = 0, then the differential equation of the form 
 
2
0 1 2 2
( ) ( ) ( ) 0
d y dy
a x a x a x y
dx dx
? ? ?