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Page 1
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Homogeneous Differential Equations and Their
Solutions
Lesson Developer: Gurudatt Rao Ambedkar
College/Department: Assistant Professor, Department of
Mathematics, Acharya Narendra Dev College, University of
Delhi
Page 2
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Homogeneous Differential Equations and Their
Solutions
Lesson Developer: Gurudatt Rao Ambedkar
College/Department: Assistant Professor, Department of
Mathematics, Acharya Narendra Dev College, University of
Delhi
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Homogeneous Differential Equations and Their Solutions
1. Learning Outcomes
2. Introduction
? 3: Linear differential equations
? 4: Second Order Linear differential equation
? 5: Homogeneous linear differential equation
? 6: Homogeneous linear differential equation with constant
coefficients
? 7: Homogeneous linear differential equation with variable
coefficients
? 8: Principal of superposition for homogeneous differential equations
? 9: Existence and Uniqueness Theorem for Linear Differential
Equations
? 10: Linearly Independent or Linearly Dependent Functions
? 11: Wronskian
? 11.1: Wronskian of solutions
? 12: Solution of Homogeneous Linear Differential Equation with
Constant Coefficient
? 12.1: Auxiliary Equation
? 12.1.1: Methods to find complementary function
? 13: Solution of Homogeneous Linear Differential Equation with
Variable Coefficient
Exercise
Summary
References
Page 3
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Homogeneous Differential Equations and Their
Solutions
Lesson Developer: Gurudatt Rao Ambedkar
College/Department: Assistant Professor, Department of
Mathematics, Acharya Narendra Dev College, University of
Delhi
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Homogeneous Differential Equations and Their Solutions
1. Learning Outcomes
2. Introduction
? 3: Linear differential equations
? 4: Second Order Linear differential equation
? 5: Homogeneous linear differential equation
? 6: Homogeneous linear differential equation with constant
coefficients
? 7: Homogeneous linear differential equation with variable
coefficients
? 8: Principal of superposition for homogeneous differential equations
? 9: Existence and Uniqueness Theorem for Linear Differential
Equations
? 10: Linearly Independent or Linearly Dependent Functions
? 11: Wronskian
? 11.1: Wronskian of solutions
? 12: Solution of Homogeneous Linear Differential Equation with
Constant Coefficient
? 12.1: Auxiliary Equation
? 12.1.1: Methods to find complementary function
? 13: Solution of Homogeneous Linear Differential Equation with
Variable Coefficient
Exercise
Summary
References
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes:
After studying this chapter you should be able to
? Understand the meaning of the term ‘Linear differential equation’
? Able to differentiate between linear differential equation with constant
coefficients and variable coefficients
? How to solve different types of linear differential equation with
constant and variable coefficients
? Understand the concept of complementary function, superposition,
wronskian, linearly dependent and linearly independent
? Understand the importance of auxiliary equation in solving of
differential equations
2. Introduction:
We face many problems in our day to day life. These problems are
sometime become too small and sometime become too serious. Everybody
wants a better future and mathematics help us to get it. We can model a life
situation with mathematics and the results of this model help us to predict
the future. The best method to develop a model is, transform the life
situation into differential equation and solve that and correlate the solution
with the problem. In this chapter we discuss about some category of
differential equations and will learn to solve them.
3. Linear Differential Equations:
A differential equation is called linear if the dependent variable and its
derivative ) ( ) ( ' y D or
dx
dy
or x y occurring in it are of the first degree and are not
multiplied together. A differential equation of the form ; B Ay
dx
dy
? ? (where A
and B are the function of x only) is called linear differential equation only.
The solution of this linear differential equation can be obtained from the
equation
) tan ( . . t cons C dx e B e y
Adx Adx
?
?
?
?
?
Where
?
Adx
e is known as integrating factor.
Page 4
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Homogeneous Differential Equations and Their
Solutions
Lesson Developer: Gurudatt Rao Ambedkar
College/Department: Assistant Professor, Department of
Mathematics, Acharya Narendra Dev College, University of
Delhi
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Homogeneous Differential Equations and Their Solutions
1. Learning Outcomes
2. Introduction
? 3: Linear differential equations
? 4: Second Order Linear differential equation
? 5: Homogeneous linear differential equation
? 6: Homogeneous linear differential equation with constant
coefficients
? 7: Homogeneous linear differential equation with variable
coefficients
? 8: Principal of superposition for homogeneous differential equations
? 9: Existence and Uniqueness Theorem for Linear Differential
Equations
? 10: Linearly Independent or Linearly Dependent Functions
? 11: Wronskian
? 11.1: Wronskian of solutions
? 12: Solution of Homogeneous Linear Differential Equation with
Constant Coefficient
? 12.1: Auxiliary Equation
? 12.1.1: Methods to find complementary function
? 13: Solution of Homogeneous Linear Differential Equation with
Variable Coefficient
Exercise
Summary
References
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes:
After studying this chapter you should be able to
? Understand the meaning of the term ‘Linear differential equation’
? Able to differentiate between linear differential equation with constant
coefficients and variable coefficients
? How to solve different types of linear differential equation with
constant and variable coefficients
? Understand the concept of complementary function, superposition,
wronskian, linearly dependent and linearly independent
? Understand the importance of auxiliary equation in solving of
differential equations
2. Introduction:
We face many problems in our day to day life. These problems are
sometime become too small and sometime become too serious. Everybody
wants a better future and mathematics help us to get it. We can model a life
situation with mathematics and the results of this model help us to predict
the future. The best method to develop a model is, transform the life
situation into differential equation and solve that and correlate the solution
with the problem. In this chapter we discuss about some category of
differential equations and will learn to solve them.
3. Linear Differential Equations:
A differential equation is called linear if the dependent variable and its
derivative ) ( ) ( ' y D or
dx
dy
or x y occurring in it are of the first degree and are not
multiplied together. A differential equation of the form ; B Ay
dx
dy
? ? (where A
and B are the function of x only) is called linear differential equation only.
The solution of this linear differential equation can be obtained from the
equation
) tan ( . . t cons C dx e B e y
Adx Adx
?
?
?
?
?
Where
?
Adx
e is known as integrating factor.
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 4
Example 1: Find the solution of the differential equation
. ) (
2
x Sin y
dx
dy
x Cos x Cos ? ?
Solution: Rewrite the given equation as
x x Sec x ySec
dx
dy
tan .
2 2
? ?
Comparing with the standard linear equation,
22
tan . A Sec x and B x Sec x ??
2
tan
( . .)
,
Adx
Sec xdx
x
Integrating factor I F e
e
e
Now the solution is
?
?
?
?
?
? ?
? ?
tan 2 tan
tan tan
tan
. tan . .
. tan 1
tan 1
xx
xx
x
ye x Sec xe dx C
ye e x C
y x Ce
?
??
? ? ?
? ? ?
?
4. Second order linear Differential Equations:
A second order differential equation is called linear if the dependent
variable y and its derivative
2
2
, ) ( ' ' ), ( '
dx
y d
dx
dy
or x y x y occurring in it are of the
first degree and are not multiplied together. A differential equation of the
form
;
2
2
D Cy
dx
dy
B
dx
y d
A ? ? ? (where C B A , , and D are the function of x only, 0 ? A ) is
called linear differential equation only.
Example2: The differential equation x y x y x Cos y e
x 1
tan ) 1 ( ' ) ( ' '
?
? ? ? ? is linear.
Value Addition: Note
We denoted the operator
dx
d
by ' or D
i.e.
. ' ' , '
2
2
2
on so and y y D
dx
y d
similary y Dy
dx
dy
? ? ? ?
Page 5
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Homogeneous Differential Equations and Their
Solutions
Lesson Developer: Gurudatt Rao Ambedkar
College/Department: Assistant Professor, Department of
Mathematics, Acharya Narendra Dev College, University of
Delhi
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Homogeneous Differential Equations and Their Solutions
1. Learning Outcomes
2. Introduction
? 3: Linear differential equations
? 4: Second Order Linear differential equation
? 5: Homogeneous linear differential equation
? 6: Homogeneous linear differential equation with constant
coefficients
? 7: Homogeneous linear differential equation with variable
coefficients
? 8: Principal of superposition for homogeneous differential equations
? 9: Existence and Uniqueness Theorem for Linear Differential
Equations
? 10: Linearly Independent or Linearly Dependent Functions
? 11: Wronskian
? 11.1: Wronskian of solutions
? 12: Solution of Homogeneous Linear Differential Equation with
Constant Coefficient
? 12.1: Auxiliary Equation
? 12.1.1: Methods to find complementary function
? 13: Solution of Homogeneous Linear Differential Equation with
Variable Coefficient
Exercise
Summary
References
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes:
After studying this chapter you should be able to
? Understand the meaning of the term ‘Linear differential equation’
? Able to differentiate between linear differential equation with constant
coefficients and variable coefficients
? How to solve different types of linear differential equation with
constant and variable coefficients
? Understand the concept of complementary function, superposition,
wronskian, linearly dependent and linearly independent
? Understand the importance of auxiliary equation in solving of
differential equations
2. Introduction:
We face many problems in our day to day life. These problems are
sometime become too small and sometime become too serious. Everybody
wants a better future and mathematics help us to get it. We can model a life
situation with mathematics and the results of this model help us to predict
the future. The best method to develop a model is, transform the life
situation into differential equation and solve that and correlate the solution
with the problem. In this chapter we discuss about some category of
differential equations and will learn to solve them.
3. Linear Differential Equations:
A differential equation is called linear if the dependent variable and its
derivative ) ( ) ( ' y D or
dx
dy
or x y occurring in it are of the first degree and are not
multiplied together. A differential equation of the form ; B Ay
dx
dy
? ? (where A
and B are the function of x only) is called linear differential equation only.
The solution of this linear differential equation can be obtained from the
equation
) tan ( . . t cons C dx e B e y
Adx Adx
?
?
?
?
?
Where
?
Adx
e is known as integrating factor.
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 4
Example 1: Find the solution of the differential equation
. ) (
2
x Sin y
dx
dy
x Cos x Cos ? ?
Solution: Rewrite the given equation as
x x Sec x ySec
dx
dy
tan .
2 2
? ?
Comparing with the standard linear equation,
22
tan . A Sec x and B x Sec x ??
2
tan
( . .)
,
Adx
Sec xdx
x
Integrating factor I F e
e
e
Now the solution is
?
?
?
?
?
? ?
? ?
tan 2 tan
tan tan
tan
. tan . .
. tan 1
tan 1
xx
xx
x
ye x Sec xe dx C
ye e x C
y x Ce
?
??
? ? ?
? ? ?
?
4. Second order linear Differential Equations:
A second order differential equation is called linear if the dependent
variable y and its derivative
2
2
, ) ( ' ' ), ( '
dx
y d
dx
dy
or x y x y occurring in it are of the
first degree and are not multiplied together. A differential equation of the
form
;
2
2
D Cy
dx
dy
B
dx
y d
A ? ? ? (where C B A , , and D are the function of x only, 0 ? A ) is
called linear differential equation only.
Example2: The differential equation x y x y x Cos y e
x 1
tan ) 1 ( ' ) ( ' '
?
? ? ? ? is linear.
Value Addition: Note
We denoted the operator
dx
d
by ' or D
i.e.
. ' ' , '
2
2
2
on so and y y D
dx
y d
similary y Dy
dx
dy
? ? ? ?
Homogeneous Differential Equations and Their Solutions
Institute of Lifelong Learning, University of Delhi pg. 5
5. Homogeneous linear Differential Equations:
A linear differential is called homogeneous linear differential equation if
the right hand side of the equation vanishes otherwise it is called non-
homogeneous.
i.e. 0
2
2
? ? ? Cy
dx
dy
B
dx
y d
A is a second order homogeneous linear differential
equation.
Example 3: 0 ' , 0 2 " ' '
2
? ? ? ? ? y xy y xy y x are homogeneous linear differential
equations.
6. Homogeneous Linear Differential Equations with constant
Coefficients-
A differential equation in which the dependent variable and the derivatives
appear only in the first degree and are not multiplied together is called a
linear differential equation.
The homogeneous linear differential equation with constant coefficient
can be written in generalize form such as
0 ...
2
2
2
1
1
1
? ? ? ? ?
?
?
?
?
y P
dx
y d
P
dx
y d
P
dx
y d
n
n
n
n
n
n
n
Where
n
P P P ....., , ,
2 1
are constants. We denoted the operator
dx
d
byD
?
1
1
..... 0
nn
n
D y PD y Py
?
? ? ? ?
Or, 0 ) ( ? y D f
Where
n
n n
P D P D y D f ? ? ? ?
?
..... ) (
1
1
acts as an operator which operate on y to
field of X.
Example 4: 0 6 3
2
2
? ? ? y
dx
dy
dx
y d
is homogeneous linear differential equation with
constant coefficients.
Read More
FAQs on Lecture 3 - Homogeneous Differential Equations and Their Solutions - Ordinary Differential Equations- Order, Degree, Formation - Engineering Mathematics
| 1. What is a homogeneous differential equation? |  |
Ans. A homogeneous differential equation is a differential equation in which all terms involving the dependent variable and its derivatives are of the same degree. In other words, the equation can be written in the form F(x, y, y', y'', ...) = 0, where F is a function that is homogeneous of degree n.
| 2. How do you solve a homogeneous differential equation? | |
Ans. To solve a homogeneous differential equation, we can use the method of separation of variables. This involves rearranging the equation so that all terms involving the dependent variable and its derivatives are on one side, and all other terms are on the other side. Then, we can integrate both sides of the equation to find the general solution.
| 3. What is the general solution of a homogeneous differential equation? | |
Ans. The general solution of a homogeneous differential equation is a solution that contains an arbitrary constant. It represents all possible solutions to the equation. To find the general solution, we integrate both sides of the equation after rearranging it in a suitable form. The resulting equation will contain an arbitrary constant, which can take on different values to represent different solutions.
| 4. Can a homogeneous differential equation have a unique solution? | |
Ans. Yes, a homogeneous differential equation can have a unique solution under certain conditions. If the equation is of first order and the initial condition (the value of the dependent variable at a specific point) is given, then the solution will be unique. However, for higher-order equations or when initial conditions are not provided, the equation may have multiple solutions or a general solution with arbitrary constants.
| 5. What are some applications of homogeneous differential equations in engineering? | |
Ans. Homogeneous differential equations have various applications in engineering. They are used to model physical phenomena such as heat conduction, fluid flow, electrical circuits, and mechanical vibrations. By solving these equations, engineers can analyze and predict the behavior of systems, design control systems, optimize performance, and solve real-world engineering problems.
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