Math 4038-1, Fall 2023: Information for Students

(Math 4038 is cross-listed as Mechanical Engineering 4563.)


1:30 -- 2:20 PM on MWF




Our class meets starting Monday, August 21, 2023. The Final Exam will be Mon., Dec. 4, at 12:30pm - 2:30pm.

Leonard Richardson

Office 386 Lockett

Office Hours

  1. MWF 11:30 AM -- 1 PM in person in my office, 386 Lockett Hall.
  2. TTh online only, at this Zoom link: Email me if you would like me to arrange a time for you on Tuesday or Thursday online.
I am available at many other times. Email first to make sure I'm able meet with you. I answer email many times daily---usually quickly.

Office phone



WebAssign and Text Book

Each student needs to click the WebAssign link and self-register for Math 4038, section 1, with Instructor Leonard Richardson, and the following class key:

lsu 7429 9452

When you register for WebAssign, please be sure to use your family name and given name in the same way that LSU has your name in Records and Registration. This is to make sure you get the correct credit for your homework. That is, the students' names must be alphabetized the same way LSU has it on class lists.

The textbook can be purchased either with or without WebAssign included. If you buy the book without WebAssign, you'll need to pay WebAssign a fee directly. If you buy the book with WebAssign you'll have a code to use so that you can show WebAssign that you have already paid their fee.

The publisher has the following package deal for LSU students in Math 4038: The ISBN for the special package – 9781284273960 ADVANCED ENGINEERING MATHEMATICS 7th Edition, by Dennis G. Zill, and published by Jones and Bartlett. This ISBN package price includes the new printed book, the student solutions manual, and WebAssign for a custom price. The campus bookstore should have the package deal available, or you can contact Customer Service at the Publisher to purchase the package deal, if it is your choice. The publisher charges less for the special ISBN package than the LSU Bookstore charges. Use this link to buy the special package for just $174 from the publisher directly: You can also buy just Webassign Access by itself for a discounted price of $53.21 directly from the publisher of the textbook. Use this link for the latter purpose:>/A>.

Be sure you obtain Edition 7 of the correct text, and also remember that there will be a charge for WebAssign if you need to buy it separately. Webassign access includes the use of the online e-book version of Zill's book, but only for the semester of this course. I think it is better for you as an engineer to have the whole book as a permanent reference on your own bookshelf, but that is just my opinion. It is not required to have a solutions manual, but I think that most students find this helpful. Just remember that you need to learn to solve the problems without the solutions manual or notes. CAUTION: If you buy a used book, you will probably need to pay WebAssign separately ($70.95). This is because the publisher should enclose a ONE_TIME registration number allowing you to use WebAssign. If the book is used however, then the enclosed number is no longer valid if the first purchaser has already used it. Bookstores prefer to sell used books because they make a much higher profit margin on used books than they do with new books. So be aware when you buy.


  1. Chapter 5: Series solutions of linear differential Equations. Bessel functions and Legendre functions.

  2. Chapter 12 -- 15: Fourier analysis, Sturm-Liouville problems and orthogonal function series. Separation of variables for partial differential equations with boundary conditions and initial conditions.

This course is cross listed with Mechanical Engineering 4563, though it is being taught only by the Mathematics Department. It is intended as a course that serves the needs of incoming graduate students in engineering who did not have such a course in preparation for graduate study, together with undergraduate Engineering majors who are preparing for graduate study. The most important goal of this course is to teach the method of separation of variables in linear partial differential equations, with applications to important equations of classical physics such as the wave equation, the heat equation, and Laplace's equation.

Many years ago, this course was preceded by a course in ordinary differential equations that included Fourier analysis. The latter course no longer exists, so it is important in Math 4038 to teach enough Fourier analysis for the treatment of partial differential equations by separation of variables. There are also important applications of this method requiring more general orthogonal function expansions, such as Fourier-Legendre and Fourier-Bessel series. However, most students in this course have not studied the use of power series to solve ordinary linear differential equations. For Bessel functions, one needs also the Frobenius method (a generalization of the power series method) for regular singular points. Thus the syllabus is organized as follows, with optional additional topics listed at the end:

Chapter 5, sections 5.1—5.3.2: Solution of ordinary linear differential equations by the method of power series, including the Frobenius method and its application to the solution of Bessel’s equation.

Chapter 12, sections 12.1—12.6.2: Orthogonal functions, Fourier series, Sturm-Liouville problems, Fourier-Bessel and Fourier-Legendre series.

Chapter 13, sections 13.1—13.8: Introduction to separation of variables, with application of the methods of Fourier expansions to boundary value problems involving the classical linear partial differential equations encountered most in applications.

Chapter 14, sections 14.1—14.3: Boundary value problems in other coordinate systems, making use of various orthogonal function expansions.

Chapter 15, sections 15.3—15.4: Introduction to the Fourier integral and Fourier transform.

Optional Additional Topics: Professors should feel free to add or replace some topics with others of comparable value, within the overall goals and purposes of this course. For example, if ones students seem to need the theorems of vector integral calculus, sections 9.12—9.16 may be appropriate, and one could include applications such as the derivation of the heat equation, for example. If there is time to teach the Fourier transform on the real line, one could solve the heat equation on the infinite line as another example.


Math 2057, plus one of the following: Math 2065, 2070, or 2090, or the equivalent.

Special Instructions during Public Health or Safety Emergencies

Your health and safety are our top priority. If you are feeling ill please contact the LSU Student Health Center for medical advice. If you are experiencing excessive stress, the Student Health Center offers mental health support as well. And please observe all the University's requirements and recommendations during emergency conditions.

We have learned from recent years that there can be unexpected changes due to the unpredictable nature of emergencies such as pandemics or extremes of weather, so that the format of the course and/or requirements may be forced to change, and if this is the case we will take care to ensure that every student is treated in a fair and considerate manner. If you have any special individual difficulty, please contact me quickly so that I can do anything possible to assist you. That said, we are looking forward to the fall semester and hoping that it will be smooth sailing.

Organization of this Class

Please understand that it is from the effort of working your way through assigned problems on paper that you learn mathematics. It is by no means sufficient to read solutions in a solutions manual! Although I hope you benefit from seeing solutions presented in class, you must not expect to learn how to solve problems just from watching. You must work out problems yourself, the hard way, in order to learn this work. Examination problems will be very similar to assigned homework problems. Thus your daily effort on homework problems will be strongly reflected in your test grades. Problems will be assigned using Webassign and they will be graded by computer. Your overall homework average converted to 100% basis at the end of the semester will replace your lowest hour test grade if that is to your advantage. If you do not understand how the computer software is grading your homework, please see me or email me so I can check online. Sometimes WebAssign is actually wrong. If you need to consult me because of a problem while using WebAssign, please email me directly at Do not use WebAssign's communication feature because it is very unreliable. I have access on WebAssign to your answers, so you do not need to email me a copy of your WebAssign answer that is causing difficulties.

Be sure to keep your written solutions to the homework problems in a notebook. Otherwise, when you review for a test or for the final exam, you won't have your work from which to study. All WebAssign retains is your answers--the work must be saved and maintained by yourself. This is obvious, but please be aware of it!


These will be closed book, in-class tests. I will provide paper and no notes or communication devices or internet may be used during a test. The problems will be similar to those in the homework. However, note that the homework is done online and graded by computer without partial credit therefore. But tests will all be graded by me and there will be partial credit, since the work is at least as important as the answer. At the end of the course, you will have the benefit of replacing your lowest hour test grade with your homework average (converted to a 100-point scale), provided that one of your hour tests is lower than your homework average.

If you miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable. Here is some General Guidance regarding appropriate reasons for absence from a test or examination. If you are in doubt, ask me as soon as possible. In any case, your lowest hour test grade can be replaced by your homework average, as explained in the homework description below.

Lateness and Classroom Conduct

Please try to arrive on time for class. But sometimes it may be unavoidable to be late. If you are late, please come right into class, doing so as quietly as you are able so as not to disturb other students. I do not want you to wait outside in the hall. You should have as much classroom time as possible, so please just come in--quietly--and take a seat even if you are late. Class time is a time for work. So when class begins please put away all cell phones, smart phones, head phones, wrist watch communicators, tablets, laptops, etc, and turn your attention to the work of the class. Thank you.


We plan to have three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your lowest hour test grade will be replaced by your homework average on a 100% scale, if it is to your advantage. I will grade your hour tests and return them to you the very next class meeting each time. Your final test average will be the sum of all your test grades divided by 5. So your final test average will be less than or equal to 100. The minimum grade for each letter grade is as follows:

A+, 97 A, 93 A-, 90 B+, 87 B, 83 B-, 80 C+, 77 C, 73 C-, 70 D+, 67 D, 63 D-, 60 F, below 60. You should save all your graded work for future study and in case you think your final grade is in error.

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It is especially important not to fall behind. It is very important to attend class and participate in homework discussions. Please do not assume you can take care of difficulties later---see me for help as soon as possible if there is something you do not understand! You are responsible for all assigned problems---not just those which we go over in class! This is a course in which many students have studied the prerequisites at other universities or several years ago. Please ask questions both in class and after class when confused or unsure. It is not possible to anticipate each student's difficulties so you need to bring them to my attention.

HOWEVER, there are businesses both locally and online that sell their own instructional services to students for fees. LSU does not authorize or approve any such services and does not recognize their competence to serve your educational interests. Advertising for any such commerical service is prohibited in our classroom. If you see such advertising being circulated in our classroom please report it to me. And buyer beware!

Computer Support

The full power of Mathematica is available on many LSU computers, including those in the Math Department's computer labs and in the Library as well. Students can access Mathematica on Tigerware through their MYLSU accounts. The DSolve command can solve many differential equations. There is a simplified Web Mathematica which is free to use online if you click on the link in this sentence. If you have not already had the Math Department's course in Mathematica, you might find it simpler to figure out how to use the Web Mathematica. It is not as versatile, but the Plot3D function is there and works fast for surface graphs. There is also a function-integrator which can be fun to use without having to learn very much about Mathematica. But do remember, this is an auxiliary resource. The time you spend working on problems on paper is the most important part of homework when it comes to learning the subject. With that understanding, Mathematica can be fun and helpful too.

Tests, Homework Assignments, and Downloads

Homework will be online using WebAssign. It will be graded on a scale of 1 point per problem or per part thereof for multi-part problems, and recorded. Please do every assigned problem on time! These assignments are not intended to be done in only one night.

At the end of the course, you will have the benefit of replacing your lowest hour test grade with your homework average (converted to a 100-point scale), provided that one of your hour tests is lower than your homework average. In Webassign, you will have 6 tries to submit an answer for each question. However, after the first 3 tries, Webassign will change the question, usually by modifying one or more coefficients appearing in the question. Different students will also receive different randomized versions of the same type of problem. Please let me know if you have difficulties using Webassign.

The 7th edition of the text is new and there might be some errors or glitches in the WebAssign grading. Please come to my office and show me if you have doubts about the correctness of WebAssign's grading. With the previous edition there were a few such errors that needed to be corrected. The computerized grading seems to be pretty good but it is not perfect. One needs to learn how to enter symbolic answers so that they will be understood. Be sure to reload this page from the website each time you visit, since the table below will be updated as the semester proceeds!

The University has clear policies requiring academic honesty. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course! Above all, never copy another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Solutions that are copied tend to be recognizable as not coming from the student, and even the precise source can be identified frequently. This means that the honorable course of action is also the practical one.


Due at Classtime on this Date

Assignments must be turned in on WebAssign in order to receive credit! Please let me know promptly at if you are having difficulties with the online assignment system or if you feel the system did not grade your work correctly. Please note that I may extend the deadlines for some homework assignments, based on how far we progress in class. This is not something that can be predicted precisely, so check regularly on WebAssign to make sure you know the corrected due dates. You can instruct WebAssign to email you automatically when a change has been made in an assignment or an assignment schedule.

August 21

Read this syllabus so you can ask questions about it in class. Obtain a copy of the text so that you are ready to begin work. If you have been away from mathematics courses for a few years, it may be necessary to review what you learned during the first three semesters of calculus. Visit the Webassign website and log in so you can see any pending homework assignments. (These will become visible beginning at Friday, August 19, 2022.) You will need to learn what symbols or formats to use when entering answers to homework problems using Webassign. Please read this Student Guide to become familiar with Webassign.

Sepember 20

Please download the example in which the method of Frobenius failed for the smaller root of the indicial equation.

September 25 On Monday, September 25, I cannot be in class with you, so I am leaving a Zoom recording with your lesson for this day. Please access this at the following link: Click Here!. In the recording, I made but fixed a few careless errors. I had difficulty figuring out how to scroll the whiteboard with the updated whiteboard program, but you can scroll ahead where I fumble and get faster to the corrections. Let me know if you have difficulties.
October 2 Please bring a list of questions to review for the first hour test, which will be on Thursday, October 4.
October 4 The first hour test will be today.

October 7

Please download Fall 2023 Test #1 with Solutions and overall class Statistics at the end.

Please download also my own full solutions to the three test problems here.

October 16

Please download (x+|x|)/2: Plot of Fourier Series on (-3pi,3pi) showing the 20th partial sum.

October 22 Please download the following file, correcting an error in WebAssign's solution to Zill, 12.2, Question 4 on the assignment. Please download Correction to Zill 12.2 Question 4 .
October 23

Please download Joseph Fourier's classic book On the Analytic Theory of Heat and my lecture notes about Sturm-Liouville Problems. The download concerning Sturm-Liouville problems is intended to help you with our course. Fourier's book is not required, but I think everyone who uses Fourier series or the Heat Equation should have, in our modern time, a free pdf copy of Fourier's classic work.

October 31

Download some supplemental files from classes taught in past years: Fourier-Bessel Series Examples and Fourier-Legendre Series Examples and also Norm and Zeros for Bessel functions and norms of Legendre functions. These downloads are not assignments. They are just supplements that add to what is in the text, including some illustrations that are better, in my opinion.

November 3

Bring review questions for the Second Hour Test, covering the work since the first hour test took place, up to and including the assignment from Sec. 12.5, due this Friday at class time.
November 6

Second Hour Test, covering the work since the first hour test took place.

November 7

Please download Fall 2023 Test #2 with Solutions and overall class Statistics at the end.

One can download an interesting discussion of a century long history of the solution to the wave equation here:

Please download D'Alembert's Solution to the Wave Equation and its history through Fourier.

November 27

Third Hour Test

November 28

Please download Test #3 in Fall 2023 with Solutions and overall class Statistics at the end.

Bring questions to review for the final exam. Remember to fill out the end of semester course evaluation form on your MyLSU page.Study for the Final Exam! This 200-point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two: 1. The grade guaranteed by the formula provided higher on this page. 2. One letter below the final exam grade. For example, if your final exam grade is A-, you are guaranteed at least a B- in the course. Thus the final exam provides a safety net that supplements the calculations specified above.

December 4

Exam Week Office Hours Monday, 10:30 AM --Noon.

December 4 The Final Exam will be Mon., Dec. 4, at 12:30pm - 2:30pm.

December 11

Please download Fall 2023 Final Examination with Solutions and overall class Statistics at the end.