Lecture 4 - Method of Undetermined Coefficients and Variation of Parameters | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Method of Undetermined Coefficients and Variation of 
Parameters 
Course Developer: Kapil Kumar and Chaman Singh 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College and A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Page 2


Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Method of Undetermined Coefficients and Variation of 
Parameters 
Course Developer: Kapil Kumar and Chaman Singh 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College and A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Page 3


Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Method of Undetermined Coefficients and Variation of 
Parameters 
Course Developer: Kapil Kumar and Chaman Singh 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College and A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Method of Undetermined Coefficients and Variation of 
Parameters 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Non-Homogeneous Linear Differential Equations 
o 3.1: Non-homogeneous Linear Differential Equation with 
Constant Coefficients  
? 4: Solutions of Non-Homogeneous Differential Equations 
? 5: Method of Undetermined Coefficients 
o 5.1: Rule to find the Particular Solution by Method of 
Undetermined Coefficients 
? Method of Variation of Parameters 
? Exercises 
? Summary 
? Reference 
 
1. Learning Outcomes: 
After reading this lesson reader will be able to understand the following 
? Non-homogeneous linear differential equation 
? Non-homogeneous linear differential equation with constant 
coefficients 
? Solutions of Non-homogeneous linear differential equations 
? Method of undetermined coefficients 
? Rule to find the Particular Solution by Method of Undetermined 
Coefficients 
? Method of Variation of Parameters 
 
 
 
 
 
Page 4


Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Method of Undetermined Coefficients and Variation of 
Parameters 
Course Developer: Kapil Kumar and Chaman Singh 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College and A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Method of Undetermined Coefficients and Variation of 
Parameters 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Non-Homogeneous Linear Differential Equations 
o 3.1: Non-homogeneous Linear Differential Equation with 
Constant Coefficients  
? 4: Solutions of Non-Homogeneous Differential Equations 
? 5: Method of Undetermined Coefficients 
o 5.1: Rule to find the Particular Solution by Method of 
Undetermined Coefficients 
? Method of Variation of Parameters 
? Exercises 
? Summary 
? Reference 
 
1. Learning Outcomes: 
After reading this lesson reader will be able to understand the following 
? Non-homogeneous linear differential equation 
? Non-homogeneous linear differential equation with constant 
coefficients 
? Solutions of Non-homogeneous linear differential equations 
? Method of undetermined coefficients 
? Rule to find the Particular Solution by Method of Undetermined 
Coefficients 
? Method of Variation of Parameters 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 4 
 
2. Introduction: 
Method of undetermined coefficients and method of variation of parameters 
are two well-known methods to solve the non-homogeneous linear 
differential equations. In this lesson, we will study about these methods and 
how to solve the non-homogeneous differential equations with the help of 
these methods. 
3. Non-Homogeneous Linear Differential Equations: 
A differential equation of the form 
 ( , , ',y'', ..., ) ( )
n
F x y y y R x ?  
or 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) ( ); a 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 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 non-homogeneous linear differential 
equation of order n if ( ) 0 Rx ? . 
If R(x) = 0, then the differential equation of the form 
 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) 0; a 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 associated homogeneous linear differential 
equation of order n with non-homogeneous linear differential equation (1). 
Value Addition: Second Order Homogeneous and Non-homogeneous 
Linear Differential Equation 
1. 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 non-homogeneous linear differential 
equation if ( ) 0 Rx ? . 
2. 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
? ? ? 
where 
0 1 2
( ), ( ), ( ) and R(x) a x a x a x are continuous functions of x only on some 
Page 5


Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Method of Undetermined Coefficients and Variation of 
Parameters 
Course Developer: Kapil Kumar and Chaman Singh 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College and A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 2 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 3 
 
Table of Contents 
 Chapter: Method of Undetermined Coefficients and Variation of 
Parameters 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Non-Homogeneous Linear Differential Equations 
o 3.1: Non-homogeneous Linear Differential Equation with 
Constant Coefficients  
? 4: Solutions of Non-Homogeneous Differential Equations 
? 5: Method of Undetermined Coefficients 
o 5.1: Rule to find the Particular Solution by Method of 
Undetermined Coefficients 
? Method of Variation of Parameters 
? Exercises 
? Summary 
? Reference 
 
1. Learning Outcomes: 
After reading this lesson reader will be able to understand the following 
? Non-homogeneous linear differential equation 
? Non-homogeneous linear differential equation with constant 
coefficients 
? Solutions of Non-homogeneous linear differential equations 
? Method of undetermined coefficients 
? Rule to find the Particular Solution by Method of Undetermined 
Coefficients 
? Method of Variation of Parameters 
 
 
 
 
 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 4 
 
2. Introduction: 
Method of undetermined coefficients and method of variation of parameters 
are two well-known methods to solve the non-homogeneous linear 
differential equations. In this lesson, we will study about these methods and 
how to solve the non-homogeneous differential equations with the help of 
these methods. 
3. Non-Homogeneous Linear Differential Equations: 
A differential equation of the form 
 ( , , ',y'', ..., ) ( )
n
F x y y y R x ?  
or 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) ( ); a 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 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 non-homogeneous linear differential 
equation of order n if ( ) 0 Rx ? . 
If R(x) = 0, then the differential equation of the form 
 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) 0; a 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 associated homogeneous linear differential 
equation of order n with non-homogeneous linear differential equation (1). 
Value Addition: Second Order Homogeneous and Non-homogeneous 
Linear Differential Equation 
1. 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 non-homogeneous linear differential 
equation if ( ) 0 Rx ? . 
2. 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
? ? ? 
where 
0 1 2
( ), ( ), ( ) and R(x) a x a x a x are continuous functions of x only on some 
Method of Undetermined Coefficients and Variation of Parameters 
Institute of Lifelong Learning, University of Delhi                                                      Pg. 5 
 
open interval I is called associated homogeneous linear differential equation 
to the non-homogeneous linear differential equation. 
 
3.1. Non-homogeneous Linear Differential Equation with Constant 
Coefficients: 
A differential equation of the form 
 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) ( ); a 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 R x
dx dx dx dx
??
? ??
? ? ? ? ? ? ? 
Where 
0 1 2 1
, , , ..., and
nn
a a a a a
?
are all constants on some open interval I is 
called non-homogeneous linear differential equation with constant 
coefficients of order n if ( ) 0 Rx ? . 
Value Addition: Note 
1. The function R(x) in the non-homogeneous differential equation 
corresponds to some external force on the system. 
2. Terms 
23
23
, , , ... ,
n
n
dy d y d y d y
dx dx dx dx
 in a differential equation may also be denoted 
by ', '', ''', ...,
n
y y y y respectively. 
 
4. Solutions of Non-Homogeneous Differential Equations: 
Consider the non-homogeneous linear differential equation of the form 
 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) ( ); a 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 R x
dx dx dx dx
??
? ??
? ? ? ? ? ? ?   (A) 
Where 
0 1 2 1
, , , ..., and
nn
a a a a a
?
are all constants on some open interval I is 
called non-homogeneous linear differential equation with constant 
coefficients of order n if ( ) 0 Rx ? . 
Then  
 
12
0 1 2 1 0 12
( ) ( ) ( ) ... ( ) ( ) 0; a 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
??
? ??
? ? ? ? ? ? ?  (B) 
is called the associated homogeneous differential equation of the non-
homogeneous differential equation (A).