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MA221: Differential Equations

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 Nth-order linear ODE with constant coefficients, you need to find the roots of an Nth-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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  • 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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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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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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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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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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  • 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)
     
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    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.
     
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  • 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” 
     
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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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  • 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.
     
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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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  • 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.
     
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  • 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.
     
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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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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.”
 
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  • 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
    1. 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.” 
       
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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.
 
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  • 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.
     
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  • 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.
     
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  • 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.”
     
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  • 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

    1. Work with exercise 2F-6.   When you finish, please check your answers to this exercise with “Section II Solutions.
       
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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.
 
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  • 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.
     
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  • 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.”
     
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  • 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.
     
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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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  • 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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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)
 
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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.
 
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  • 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.
     
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  • 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.
     
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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.
 
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  • 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.
     
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  • 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.
     
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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.
 
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  • 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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