Unit 3: Higher-Order Linear ODEs While one can always convert a higher-order ODE into a system of first-order ODEs, this is not always the best approach toward solving the ODE. Homogeneous linear ODEs with constant coefficients are a particularly easy-to-solve class of ODEs. In order to find a solution of an N^{th}-order linear ODE with constant coefficients, you need to find the roots of an N^{th}-order polynomial.
Finding solutions is not quite as simple for the case of linear ODEs with variable coefficients—that is, with coefficients that are functions of the independent variable. These ODEs are not generally solvable in closed form, but methods for solving some special cases do exist.
The general solution to non-homogeneous linear ODEs is the addition of the general solution of the related homogeneous ODE plus a particular solution of the non-homogeneous ODE. There are a number of approaches to finding a particular solution, and we will sample several of the most straightforward. However, finding particular solutions is less a process of following an algorithm than it is an art form.
You are already familiar with the idea that nearly any “well-behaved” function can be approximated by a power series. You can use this concept to find solutions to a wide range of ODEs. In most cases, the solution itself is expressed as a power series, a fact that has led to the development of an enormous field of applied mathematics known as “special functions theory,” to which we will return later. In this unit, we will examine the solution of a specific example—Bessel’s equation (which was actually introduced by Bernoulli!).
Unit3 Learning Outcomes
Upon successful completion of this unit, the student should be able
to:
- Find the solution for second-order ordinary differential equations.
- Find the solution for second-order ordinary differential equations
within applications involving Newton’s law of motion.
- Find the solution for second-order differential equations within
applications involving spring-mass systems.
- Find the solution for second-order ordinary differential equations
within applications involving air resistance.
- Find the solution for second-order ordinary differential equations
within applications involving Schrödinger’s one-dimensional
time-independent equation.
- Find the solution for higher order differential equations.
- Find the solution for homogeneous ordinary differential equations
with constant coefficients.
- Find the solution for homogeneous ordinary differential equations
with variable coefficients.
- Find the solution for Euler-Cauchy ordinary differential equations.
- Find the particular solution for non-homogeneous ordinary
differential equations.
- Apply the use of linear differential operators within selected
applications.
- Find the solution for an ordinary differential equation by the
method of undetermined coefficients.
- Identify the trial functions which arise within solutions involving
the method of undetermined coefficients.
- Find the solution for an ordinary differential equation using the
variation of parameters.
- Distinguish between the use of the method of undetermined
coefficients and variation of parameters to solve ordinary
differential equations.
- Find the solution for an ordinary differential equation using a
method called the reduction of order.
- Use a method called the reduction of order to convert an ordinary
differential equation to another ordinary differential equation of
lower order.
- Find the solution for an ordinary differential equation using the
method of inverse operators.
- Identify the inverse operators which arise within solutions
involving the method of inverse operators.
- Find the power series solution for an ordinary differential
equation.
- Distinguish between a power series solution for an ordinary
differential equation about an ordinary point and a singular point.
- Find the solution for an ordinary differential equation using the
Frobenius method.
- Describe how the general solution depends upon the order of the
Bessel equation.
3.1 Examples of Second-Order Linear Differential Equations
3.1.1 Newton’s Law of Motion
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Basic
Concepts”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read only the section titled “Differential
Equation” on page 2.
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displayed on the webpages above.
- Reading: NYU: Mark Tuckerman’s: “G25.2600: Molecular Dynamics”:
“Notes for Lecture 1”: “Newton’s Laws of Motion”
Link: NYU: Mark Tuckerman’s: “G25.2600: Molecular Dynamics”: “Notes
for Lecture 1”: “Newton’s Laws of
Motion”
(HTML)
Instructions: Click on link above and read the notes included.
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3.1.2 Motion of a Mass on a Spring
- Web Media: YouTube: Jason Gregersen’s “Modeling Spring Motion
Using Differential Equations Part One”
Link: YouTube: Jason Gregersen’s “Modeling Spring Motion Using
Differential Equations Part
One” (YouTube)
Instructions: Click on the link above to watch this video (5:11
minutes), which models spring motion (assuming that there is no air
resistance).
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displayed on the webpages above.
3.1.3 Motion with Air Resistance
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Mechanical Vibrations”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Mechanical
Vibrations”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Go to page 162 and read the section titled
“Mechanical Vibrations,” (pages 162-164).
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displayed on the webpages above.
3.1.4 One-Dimensional Time-Independent Schrödinger Equation
- Reading: Georgia Institute of Technology: C. David Sherrill’s A
Brief Review of Elementary Quantum Chemistry: “The Schrödinger
Equation”: “The Time-Independent Schrödinger Equation”
Link: Georgia Institute of Technology: C. David Sherrill’s A Brief
Review of Elementary Quantum Chemistry: “The Schrödinger Equation”:
“The Time-Independent Schrödinger
Equation”
(HTML)
Instructions: Click on the link above and read the entire
section.
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displayed on the webpages above.
3.2 Higher Order Differential Equations
- Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and
Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators
and Higher Order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises:
“Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher
Order
ODE’s”
(PDF)
Instructions: Please click on the link above and go to page 6.
Work with exercise 2F-1 (items c, d, e, and f,); exercise 2F-2; and
exercise 2F-3, item d. When you finish, please check your answers
with “Section II
Solutions”
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displayed on the webpages above.
3.2.1 Homogeneous Linear ODEs with Constant Coefficients
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Second Order Differential
Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Second Order Differential
Equations”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Go to Page 104 and read the sections
titled: “Basic Concepts,” “Real, Distinct Roots,” “Complex Roots,”
and “Repeated Roots,” pages 104-121. Then go to page 126 and read
the sections titled “Fundamental Sets of Solutions” and “More on the
Wronskian” (pages 126-136).
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displayed on the webpages above.
Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Higher Order Linear ODE’s”: “2.3 Higher Order Linear ODE’s” Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Higher Order Linear ODE’s”: “2.3 Higher Order Linear ODE’s” (PDF)
Also available in:
HTMLEPUB
Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 57. Please read the section titled “2.3 Higher Order Linear ODE’s” (pages 57-60).Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Jiri Lebl, and the original version can be found here (PDF).
Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Second Order (Homogeneous)” Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Second Order (Homogenous)” (PDF)
Instructions: Click on the link above, then scroll down until you find the tutorial titled “Second Order (Homogeneous).” Click on the corresponding link. Go to page 5 and work on exercises 1 to 16 on pages 5 to 7. After finishing each exercise, click on the “EXERCISE” link for full worked solution.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” (PDF)
Instructions: Please click on the link above and go to page 3. Work with exercises 2C-1 and 2C-2. When you finish, please check your answers with “Section II Solutions”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.2.2 Homogeneous Linear ODEs with Variable Coefficients
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Reduction of Order”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Reduction of
Order”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Go to page 122 and read the section titled
“Reduction of Order” (pages 122-125).
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displayed on the webpages above.
- Assessment: Caltech: Sean Mauch’s Introduction to Methods of
Applied Mathematics: “Ordinary Differential Equations”: “Chapter
17”: “17.7 Additional Exercises”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied
Mathematics: “Ordinary Differential Equations”: “Chapter
17”: “17.7 Additional
Exercises” (PDF)
Instructions: Please click on the link above and go to page 955. Work on exercises 17.18, 17.19, 17.20, and 17.21. After finishing each exercise, click on the “solution” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.2.3 Euler-Cauchy ODEs
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Series Solutions to Differential Equations”: “Euler Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Series Solutions to Differential Equations”: “Euler
Equations”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Go to page 340 and read the section titled:
“Euler Equations” (pages 340-344).
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displayed on the webpages above.
Activity: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 5” Link: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 5” (PDF)
Instructioons: Click on the link above and scroll down until you see “M344 Advanced Engineering Math.” Then click on the link titled “Lecture 5: Cauchy-Euler Equations, Method of Frobenius.” Go to page 4 and look for “Practice Problems.” Complete items a and b under problem 2 and then check for the answers.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises” Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises” (PDF)
Instructions: Click on the link above. Click on “PDF (Portable Document Format).” Go to page 953. Work on exercises 17.11, 17.12, 17.13, and 17.14. After finishing each exercise, click on the “solution” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.2.4 Non-homogeneous Linear ODEs – Finding Particular Solutions
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Nonhomogeneous Differential
Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Nonhomogeneous Differential
Equations”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read the section titled “Nonhomogeneous
Differential Equations” on pages 137-138.
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displayed on the webpages above.
3.2.4.1 Linear Differential Operators
- Reading: MIT: Professor Arthur Mattuck’s 18.03 Notes and
Exercises: “Notes”: “Linear Differential Operators”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises:
“Notes”: “Linear Differential
Operators”
(PDF)
Instructions: Please click on the link above and read pages 1 to 5
of “Linear Differential Operators.”
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and
Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators
and Higher Order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises:
“Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and
Higher Order
ODE’s”
(PDF)
Instructions: Please click on the PDF linked above and go to page- Work with exercise 2F-1 items “a” and “b”, and exercise 2F-3
items “a” and “b”. When you finish, please check your answers for
these exercises with “Section II
Solutions.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
- Work with exercise 2F-1 items “a” and “b”, and exercise 2F-3
items “a” and “b”. When you finish, please check your answers for
these exercises with “Section II
Solutions.”
3.2.4.2 Trial Solutions – Method of Undetermined Coefficients
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Undetermined Coefficients”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Undetermined
Coefficients”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read the section titled “Undetermined
Coefficients” on pages 139-155.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Undetermined Coefficients” Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Undetermined Coefficients” (PDF)
Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Undetermined Coefficients” on pages 355-356.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: Graham McDonald’s “Second Order (Inhomogeneous)” Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: Graham McDonald’s “Second Order (Inhomogenous)” (PDF)
Instructions: Click on the link above, then scroll down until you find the tutorial titled “Second Order (Inhomogeneous).” Click on the corresponding link. Go to page 5 and work on exercises 1 to 13 on pages 5 and 6. After finishing each exercise, click on the “EXERCISE” link for full worked solutions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” (PDF)
Instructions: Please click on the PDF linked above and go to page 3. Work with exercises 2C-7 and 2C-8 on page 3. When you finish, please check your answers for these exercises with “Section II Solutions.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s” (PDF)
Instructions: Please click on the PDF linked above and go to page- Work with exercise 2F-6. When you finish, please check your
answers to this exercise with “Section II
Solutions.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
- Work with exercise 2F-6. When you finish, please check your
answers to this exercise with “Section II
Solutions.”
3.2.4.3 Variation of Parameters
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Variation of Parameters”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Second Order Differential Equations”: “Variation of
Parameters”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read the section titled “Variation of
Parameters” on pages 156-161.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Variation of Parameters” Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Variation of Parameters” (PDF)
Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled: “Variation of Parameters” on pages 357-362.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2D. Variation of Parameters” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2D. Variation of Parameters” (PDF)
Instructions: Please click on the PDF linked above and go to page 4. Work with exercises 2D-1 and 2D-2 on page 4. When you finish, please check your answers to these exercises with “Section II Solutions.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 21”: “21.10 Exercises” Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 21”: “21.10 Exercises” (PDF)
Instructions: Click on the link above. Click on “PDF (Portable Document Format).” Go to page 1117 and work on exercises 21.3, 21.4, and 21.5. After finishing each exercise, click on the “solution” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.2.4.4 Reduction of Order
- Reading: CliffsNotes: “Reduction of Order”
Link: CliffsNotes: “Reduction of
Order”
(HTML)
Instructions: Read this webpage.
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displayed on the webpages above.
- Reading: Efunda’s “Higher Order Linear Differential Equations”:
“Method of Reduction of Order”
Link: Efunda’s “Higher Order Linear Differential Equations”:
“Method of Reduction of
Order”
(HTML)
Instruction: Please click on the above link and read the entire article.
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3.2.4.5 Method of Inverse Operators
- Reading: Efunda’s “Higher Order Linear Differential Equations”:
“Inverse Operators”
Link: Efunda’s “Higher Order Linear Differential Equations”:
“Inverse
Operators”
(HTML)
Instruction: Please click on the above link and read the entire
section.
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displayed on the webpages above.
3.3 Power Series Solutions of Linear Differential Equations
- Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Power Series
Methods”: “7.1 Power Series”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Power Series
Methods”: “7.1 Power
Series”
(PDF)
Also available in:
HTML
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA221-book-Notes-on-Diffy-Qs-Jiri-Lebl.epub)
Instructions: Click on the link above and then click on the link
that says “Download the book as PDF.” Go to page 261. Please read
the section titled “7.1 Power Series”
on pages 261-268.
Terms of Use: The work above is released under a [Creative Commons
Attribution-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/us/) (HTML).
It is attributed to Jiri Lebl, and the original version can be
found [here](http://www.jirka.org/diffyqs/) (PDF).
3.3.1 Power Series Solutions about an Ordinary Point
- Reading: Lamar University: Paul Dawkins’ Differential Equations:
“Series Solutions to Differential Equations”: “Series Solutions to
Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations:
“Series Solutions to Differential Equations”: “Series Solutions to
Differential
Equations”
(PDF)
Instructions: Click on the link above and then look for the line
that says “Here is the file you requested: Differential Equations
(Math 3301).” Click on the link associated with “Differential
Equations (Math 3301).” Read the section titled “Series Solutions
to Differential Equations” on pages 330-339.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Series Solutions” Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Series Solutions” (PDF)
Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Series Solutions” on pages 370-373.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Power Series: 6C. Solving Second-order ODE’s” Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Power Series: 6C. Solving Second-order ODE’s” (PDF)
Instructions: Go to page 2 of the PDF. Do exercises 6C-2, 6C-3, 6C-4, 6C-5, 6C-6 and 6C-7. When you finish, check your work with the Solutions.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.3.2 Singular Points: The Frobenius Method
- Reading: California State University, East Bay: Massoud Malek’s
Differential Equations: “Series Solutions of Linear Differential
Equations”
Link: California State University, East Bay: Massoud Malek’s
Differential Equations: “Series Solutions of Linear Differential
Equations” (PDF)
Instructions: Click on the link above. Scroll down until you find
“Series Solutions of LDE.” Click on that link and read the entire
document.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
Reading: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.2 Regular Singular Points of Second Order Equations” Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.2 Regular Singular Points of Second Order Equations” (PDF)
Instructions: Click on the link above and then click on “PDF (Portable Document Format).” Go to page 1198. Read the section titled “23.2 Regular Singular Points of Second Order Equations” on pages 1198-1215.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.5 Exercises” Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.5 Exercises” (PDF)
Instructions: Click on the link above and then click on “PDF (Portable Document Format).” Go to page 1220 and work on exercises 23.3, 23.5, 23.6, and 23.7. After finishing each exercise, click on the “solution” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
3.3.3 Bessel’s Equation
- Reading: University of Hartford: Virginia Noonburg’s M344 Advanced
Engineering Mathematics: “Lecture 6: Bessel’s Equation”
Link: University of Hartford: Virginia Noonburg’s M344 Advanced
Engineering Mathematics: “Lecture 6: Bessel’s
Equation” (PDF)
Instructions: Click on the link above and scroll down until you see
“M344 Advanced Engineering Math.” Under that title, click on the
link that reads: “Lecture 6: Bessel’s Equation.” Read the entire
lecture.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Holon Institute of Technology: Professor Benzion
Shklyar’s “Solution of Bessel Equation with v=1/3”
Link: Holon Institute of Technology: Professor Benzion Shklyar’s
“Solution of Bessel Equation with
v=1/3”
(PDF)
Instructions: Click on the link above and scroll down until you find “Bessel Equation.” Under that title, click on the link for “Solution of Bessel Equation with v=1/3.” Read the question and work on it on your own. After you finish, look under the question for all the steps of the solution.
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