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Page 1
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Paper: Ordinary Differential Equations
Lesson: Series Solution of Differential Equation
Course Developer: Brijendra Yadav
Departmental / College: Department of Mathematics,
Acharya Narendra Dev College, University of Delhi
Page 2
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Paper: Ordinary Differential Equations
Lesson: Series Solution of Differential Equation
Course Developer: Brijendra Yadav
Departmental / College: Department of Mathematics,
Acharya Narendra Dev College, University of Delhi
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Series Solutions of Differential Equations
1. Learning Outcomes
2. Introductions
3. Some basic definitions (power series, Analytic Function)
3.1 Power Series
3.2 Convergence of a power series
3.3 Analytic Function
3.4 Ordinary Point
3.5 Singular Point
3.6 Regular singular Point
3.7 Irregular singular Point
4. Power series salutation
4.1 The power series solution in powers of
0
() xx ?
4.2 Initial Value Problem
4.3 Frobeinus method
4.4 Indicial equation, Indicating the form of solutions
4.5 Frobenius method Unequal roots (not differing by an integer)
4.6 Frobenius method for double roots
4.7 Frobenius method Unequal roots (but solution is not linearly
independent)
Exercises
Summary
References
Page 3
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Paper: Ordinary Differential Equations
Lesson: Series Solution of Differential Equation
Course Developer: Brijendra Yadav
Departmental / College: Department of Mathematics,
Acharya Narendra Dev College, University of Delhi
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Series Solutions of Differential Equations
1. Learning Outcomes
2. Introductions
3. Some basic definitions (power series, Analytic Function)
3.1 Power Series
3.2 Convergence of a power series
3.3 Analytic Function
3.4 Ordinary Point
3.5 Singular Point
3.6 Regular singular Point
3.7 Irregular singular Point
4. Power series salutation
4.1 The power series solution in powers of
0
() xx ?
4.2 Initial Value Problem
4.3 Frobeinus method
4.4 Indicial equation, Indicating the form of solutions
4.5 Frobenius method Unequal roots (not differing by an integer)
4.6 Frobenius method for double roots
4.7 Frobenius method Unequal roots (but solution is not linearly
independent)
Exercises
Summary
References
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes : After reading this lesson, the read will be able
to understand
? Power series expansion about
0
xx ?
? Analytic nature of () Px and () Qx at
0
xx ?
? Nature of equations when
0
xx ? is an ordinary point
0
xx ? is a singular point
0
xx ? is a regular singular point
0
xx ? is a irregular singular point
? Solution of differential equation by Frobenius method.
? Forbenius method when 0 x ? is a regular singular point.
? Formation of indicial equation of differential equations.
? Power series solution when roots are distinct (Frobenius Method)
? Power series solution when roots are equal (Frobenius Method)
? Method of Power series solution when two solution (
1
() yx and
2
() yx ) are not linearly independent.
2. Introduction:
If a homogeneous linear differential equation has constant coefficients, it can
be solved by algebraic method, and its solutions are elementary functions,
known from calculus ( ,cos ,sin .)
x
e x xetc However, if such an equation has
variable coefficients (e.g., function ofx ), it must usually be solved by other
methods. Legendre’s equation
2
(1 ) " 2 ' ( 1) 0 x y xy n n y ? ? ? ? ? and Bessel’s
equation
2 2 2
( " ' ( ) 0) x y xy x n y ? ? ? ? are very important equations of this type.
These and other equations and their solution play a basic role in applied
mathematics. In this chapter we shall discuss the two standard methods of
solution and their applications: The power series method which fields
solutions in the form power series, and an extension of it, called the
Frobenius method.
Page 4
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Paper: Ordinary Differential Equations
Lesson: Series Solution of Differential Equation
Course Developer: Brijendra Yadav
Departmental / College: Department of Mathematics,
Acharya Narendra Dev College, University of Delhi
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Series Solutions of Differential Equations
1. Learning Outcomes
2. Introductions
3. Some basic definitions (power series, Analytic Function)
3.1 Power Series
3.2 Convergence of a power series
3.3 Analytic Function
3.4 Ordinary Point
3.5 Singular Point
3.6 Regular singular Point
3.7 Irregular singular Point
4. Power series salutation
4.1 The power series solution in powers of
0
() xx ?
4.2 Initial Value Problem
4.3 Frobeinus method
4.4 Indicial equation, Indicating the form of solutions
4.5 Frobenius method Unequal roots (not differing by an integer)
4.6 Frobenius method for double roots
4.7 Frobenius method Unequal roots (but solution is not linearly
independent)
Exercises
Summary
References
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes : After reading this lesson, the read will be able
to understand
? Power series expansion about
0
xx ?
? Analytic nature of () Px and () Qx at
0
xx ?
? Nature of equations when
0
xx ? is an ordinary point
0
xx ? is a singular point
0
xx ? is a regular singular point
0
xx ? is a irregular singular point
? Solution of differential equation by Frobenius method.
? Forbenius method when 0 x ? is a regular singular point.
? Formation of indicial equation of differential equations.
? Power series solution when roots are distinct (Frobenius Method)
? Power series solution when roots are equal (Frobenius Method)
? Method of Power series solution when two solution (
1
() yx and
2
() yx ) are not linearly independent.
2. Introduction:
If a homogeneous linear differential equation has constant coefficients, it can
be solved by algebraic method, and its solutions are elementary functions,
known from calculus ( ,cos ,sin .)
x
e x xetc However, if such an equation has
variable coefficients (e.g., function ofx ), it must usually be solved by other
methods. Legendre’s equation
2
(1 ) " 2 ' ( 1) 0 x y xy n n y ? ? ? ? ? and Bessel’s
equation
2 2 2
( " ' ( ) 0) x y xy x n y ? ? ? ? are very important equations of this type.
These and other equations and their solution play a basic role in applied
mathematics. In this chapter we shall discuss the two standard methods of
solution and their applications: The power series method which fields
solutions in the form power series, and an extension of it, called the
Frobenius method.
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
3. Some Basic Definitions:
3.1. Power Series:
The power series method is the standard basic method for solving linear
differential equations.
An infinite series of the form
2
0 0 1 0 2 0
0
( ) ( ) ( ) ...
n
n
n
c x x c c x x c x x
?
?
? ? ? ? ? ? ?
?
(1)
is called a power series in
0
( ), xx ? where
0 1 2 3
, , , ... c c c c are constants called
coefficients of the series and x is a variable.
In Particular, a Power series in x is an infinite series
23
0 1 2 3
0
...
n
n
n
c x c c x c x c x
?
?
? ? ? ? ?
?
(2)
For example the exponential function
x
e has the power series
2
345
0
1
1 ...
3! 4! 5!
! 2!
n
x
n
xx
x x x
ex
n
?
?
?
? ? ? ? ? ? ? ?
?
Similarly the other examples of the power series are
2
0
1
1 ..., 1
1
n
n
x x x x
x
?
?
? ? ? ? ? ?
?
?
(geometric series)
2 2 4
0
( 1)
cos 1 ...
(2 )! 2! 4!
nn
n
x x x
x
n
?
?
?
? ? ? ? ? ?
?
2 1 3 5
0
( 1)
sin ...
(2 1)! 3! 5!
nn
n
x x x
xx
n
? ?
?
?
? ? ? ? ? ?
?
?
Page 5
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Paper: Ordinary Differential Equations
Lesson: Series Solution of Differential Equation
Course Developer: Brijendra Yadav
Departmental / College: Department of Mathematics,
Acharya Narendra Dev College, University of Delhi
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Series Solutions of Differential Equations
1. Learning Outcomes
2. Introductions
3. Some basic definitions (power series, Analytic Function)
3.1 Power Series
3.2 Convergence of a power series
3.3 Analytic Function
3.4 Ordinary Point
3.5 Singular Point
3.6 Regular singular Point
3.7 Irregular singular Point
4. Power series salutation
4.1 The power series solution in powers of
0
() xx ?
4.2 Initial Value Problem
4.3 Frobeinus method
4.4 Indicial equation, Indicating the form of solutions
4.5 Frobenius method Unequal roots (not differing by an integer)
4.6 Frobenius method for double roots
4.7 Frobenius method Unequal roots (but solution is not linearly
independent)
Exercises
Summary
References
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning outcomes : After reading this lesson, the read will be able
to understand
? Power series expansion about
0
xx ?
? Analytic nature of () Px and () Qx at
0
xx ?
? Nature of equations when
0
xx ? is an ordinary point
0
xx ? is a singular point
0
xx ? is a regular singular point
0
xx ? is a irregular singular point
? Solution of differential equation by Frobenius method.
? Forbenius method when 0 x ? is a regular singular point.
? Formation of indicial equation of differential equations.
? Power series solution when roots are distinct (Frobenius Method)
? Power series solution when roots are equal (Frobenius Method)
? Method of Power series solution when two solution (
1
() yx and
2
() yx ) are not linearly independent.
2. Introduction:
If a homogeneous linear differential equation has constant coefficients, it can
be solved by algebraic method, and its solutions are elementary functions,
known from calculus ( ,cos ,sin .)
x
e x xetc However, if such an equation has
variable coefficients (e.g., function ofx ), it must usually be solved by other
methods. Legendre’s equation
2
(1 ) " 2 ' ( 1) 0 x y xy n n y ? ? ? ? ? and Bessel’s
equation
2 2 2
( " ' ( ) 0) x y xy x n y ? ? ? ? are very important equations of this type.
These and other equations and their solution play a basic role in applied
mathematics. In this chapter we shall discuss the two standard methods of
solution and their applications: The power series method which fields
solutions in the form power series, and an extension of it, called the
Frobenius method.
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
3. Some Basic Definitions:
3.1. Power Series:
The power series method is the standard basic method for solving linear
differential equations.
An infinite series of the form
2
0 0 1 0 2 0
0
( ) ( ) ( ) ...
n
n
n
c x x c c x x c x x
?
?
? ? ? ? ? ? ?
?
(1)
is called a power series in
0
( ), xx ? where
0 1 2 3
, , , ... c c c c are constants called
coefficients of the series and x is a variable.
In Particular, a Power series in x is an infinite series
23
0 1 2 3
0
...
n
n
n
c x c c x c x c x
?
?
? ? ? ? ?
?
(2)
For example the exponential function
x
e has the power series
2
345
0
1
1 ...
3! 4! 5!
! 2!
n
x
n
xx
x x x
ex
n
?
?
?
? ? ? ? ? ? ? ?
?
Similarly the other examples of the power series are
2
0
1
1 ..., 1
1
n
n
x x x x
x
?
?
? ? ? ? ? ?
?
?
(geometric series)
2 2 4
0
( 1)
cos 1 ...
(2 )! 2! 4!
nn
n
x x x
x
n
?
?
?
? ? ? ? ? ?
?
2 1 3 5
0
( 1)
sin ...
(2 1)! 3! 5!
nn
n
x x x
xx
n
? ?
?
?
? ? ? ? ? ?
?
?
Series Solution of Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 5
3.2. Convergence of a power series:
The power series
2
0 0 1 0 2 0
0
( ) ( ) ( ) ...
n
n
n
c x x c c x x c x x
?
?
? ? ? ? ? ? ?
?
Converges (absolutely) for
0
x x R ??
where
1
lim ,
n
n
n
c
R
c
??
?
? provided the limit exits. (3)
R is called the radius of convergence of the power series (1) in the interval
( , ) RR ? which is said to be the interval of convergence.
If R ?? for a power series, then the interval of convergence of that power
series will be ( , ) ? ? ? i.e., the real line.
From the above condition we shall use the following results:
a. A power series represents a continuous function with its interval of
convergence
b. A power series can be differentiated term wise with in interval of
convergence.
3.3. Analytic Function:
A function () fx defined on an interval containing the point
0
xx ? is called
analytic at
0
x if its Taylor series
0
0
()
()
!
n
n
no
fx
xx
n
?
?
?
?
(4)
exists and converges to () fx for all x in the interval of convergence of ( 4 ).
Read More
FAQs on Lecture 4 - Series Solution of Differential Equation - Ordinary Differential Equations- Order, Degree, Formation - Engineering Mathematics
| 1. What is a series solution of a differential equation? |  |
Ans. A series solution of a differential equation is a method used to find an approximate solution by representing the unknown function as an infinite series. This series is then substituted into the differential equation, resulting in a recurrence relation for the coefficients of the series. By solving this recurrence relation, we can determine the values of the coefficients and obtain an approximate solution to the differential equation.
| 2. When is the series solution method useful in solving differential equations? | |
Ans. The series solution method is particularly useful when the differential equation cannot be easily solved using other techniques such as separation of variables or integrating factors. It is also helpful when the desired solution is in the form of a power series or when the coefficients of the differential equation are given as power series.
| 3. How is a series solution obtained for a differential equation? | |
Ans. To obtain a series solution for a differential equation, we assume that the unknown function can be expressed as a power series. We substitute this power series into the differential equation and equate the coefficients of each power of the independent variable. This results in a recurrence relation for the coefficients, which can be solved to find their values. By substituting these values back into the power series, we obtain the series solution.
| 4. Is the series solution always an exact solution to the differential equation? | |
Ans. No, the series solution obtained using this method is usually an approximate solution rather than an exact one. The accuracy of the series solution depends on the convergence properties of the power series. If the power series converges to the desired solution, the series solution will be closer to the exact solution. However, in some cases, the power series may not converge or may converge to a different function, leading to a less accurate approximation.
| 5. What are some applications of series solutions in engineering? | |
Ans. Series solutions are commonly used in engineering to solve differential equations that arise in various fields. They are particularly useful in analyzing and modeling physical systems, such as heat conduction, fluid flow, and electrical circuits. Series solutions also play a significant role in solving boundary value problems, eigenvalue problems, and problems involving complex variables. By obtaining approximate solutions using series methods, engineers can gain insights into the behavior of these systems and make informed design decisions.
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