What to cover in a differential equations module?

[matqkks]

I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.

I want to write a course that it motivates students and has an impact. What topics and what is the most motivating way to introduce differential equations? I want a well-structured and practical approach to differential equations. It is for mathematics and physics students.

 

[gleem]

Disclaimer: I am a physicist and have never taught a diff Eq course although I have taken several. But here are my thoughts based on what I can remember from those courses.

A good course takes time to develop. It is doubtful that your first iteration will be what you are seeking.

I assume you will be using a text. I would select a text that best reflects the goals for your course. You can delete or augment various topics as you see fit. Is it only ODE or Include PDE?

What is your problem with Laplace transforms? It would seem beneficial to introduce them.

Since it is a mix of math and physics majors it would seem to me to use an approach that favors the math majors. The physics majors should benefit. Showing the wide application of differential equations including economics, biology, dynamic systems, epidemiology as well as the usual physics and engineering apps. should help inspire an interest in the subject and may help students find a field of application to pursue later on.

 

[matqkks]

Disclaimer: I am a physicist and have never taught a diff Eq course although I have taken several. But here are my thoughts based on what I can remember from those courses.

A good course takes time to develop. It is doubtful that your first iteration will be what you are seeking.

I assume you will be using a text. I would select a text that best reflects the goals for your course. You can delete or augment various topics as you see fit. Is it only ODE or Include PDE?

What is your problem with Laplace transforms? It would seem beneficial to introduce them.

Since it is a mix of math and physics majors it would seem to me to use an approach that favors the math majors. The physics majors should benefit. Showing the wide application of differential equations including economics, biology, dynamic systems, epidemiology as well as the usual physics and engineering apps. should help inspire an interest in the subject and may help students find a field of application to pursue later on.

Only ODE. In general when I solve des it is easier to use the normal methods of differential equations. It has got to be of the correct form in order to take the inverse LT

 

[fresh_42]

I want to write a course that it motivates students and has an impact.

I would elaborate on the meaning of the exponential function in differential equations / geometry. It is the anti-derivative per se. Why is this the case?

Next things are vector fields. Teach to draw them given the original equations and then explain what flows are and how they represent solutions.

And, of course, Picard Lindelöf is probably a must have.

Besides the technical algorithms to solve ODE you could have a few useful applications: Lotka-Volterra, SIR models, heat equation.

 

[Mark44]

Unlike @gleem, I have taught classes in differential equations many times.

What topics and what is the most motivating way to introduce differential equations? I want a well-structured and practical approach to differential equations. For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.

Why not? This is a useful technique to solve initial condition problems.

I want to write a course that it motivates students and has an impact.

Writing a whole course seems like a lot of work, assuming that you're not going to use a textbook. OTOH, there are lots of textbooks to choose from. A couple that come to mind are Boyce & Di Prima or Gilbert Strang. There are many others. These textbooks contain applied examples that can serve to motivate the students. Once you decide on the textbook you want to use, you can choose the sections that you want to cover in your course.

It has got to be of the correct form in order to take the inverse LT

There are a number of algebraic techniques that can be used so as to get the final equation in a form that tables of inverse Laplace transforms can be used.

 

[vela]

In general when I solve des it is easier to use the normal methods of differential equations. It has got to be of the correct form in order to take the inverse LT

Like you, Laplace isn't the method I'd typically resort to for solving differential equations, but I think students should at least be introduced to the concept of transforms as well as seeing the LT as a change of basis. You don't have to spend a lot of time on the topic. You can focus on fairly basic applications of the Laplace transform. I wouldn't bother with the properties like time shifting or with the convolution theorem. Those details can be left for later courses.

 

[gleem]

Again since there are math majors introducing Laplace transforms should be considered. The book on ODEs by W. A. Adkins and M. G. Davidson makes a case for introducing LT early.

https://www.amazon.com/dp/1461436176/?tag=pfamazon01-20

 

[fresh_42]

I'd like to state a minority opinion. I think LT is part of complex calculus and should be taught there. Instead, the calculus of variations would historically make more sense.

And before I get stoned, this is my personal opinion.

 

[berkeman]

And before I get stoned,

That has a couple of different meanings...

 

[Mondayman]

You could include both topics. Get two birds stoned at once.

 

[gleem]

I think LT is part of complex calculus and should be taught there.

Yes, it should properly be discussed there.

I learned to use LT 60 years ago in my first ODE course. It was introduced ad hoc. If the OP did not adamantly state that he would not teach it I certainly would not have made any mention to include it. It is my assumption that just like series solutions and numerical methods LT is by now always included in an ODE course.

And before I get stoned, this is my personal opinion.

Since I am not without sin I cannot cast a stone.

 

[Mark44]

And before I get stoned, this is my personal opinion.

After one is stoned, personal opinions are often discounted.

 

[Mondayman]

After one is stoned, personal opinions are often discounted.

 

[jtbell]

in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.

You might not use them, but professors who teach later courses in which your students will be using differential equations, might expect their students to know Laplace transforms.

In the US at least, I would find out from other professors in the math department how the course has been taught in the past, and what expectations (if any) need to be satisfied for later courses that this course "feeds into". The official course description that is given online and in the school's official catalog may include a list of areas or topics that the course is supposed to cover. Different countries might have different customs in this regard, of course.

 

[George Jones]

I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.

I want to write a course that it motivates students and has an impact. What topics and what is the most motivating way to introduce differential equations? I want a well-structured and practical approach to differential equations. It is for mathematics and physics students.

The topics that can be included is partially determined by the the number of lecture hours in the course.

A nice, somewhat lively, text is "Differential Equations and Their Applications: An Introduction to Applied Mathematics" by Braun (recommended by mathwonk). It includes nice motivation for Laplace transforms.

That has a couple of different meanings...

One of which is legal in Canada.

 

[vanhees71]

Only ODE. In general when I solve des it is easier to use the normal methods of differential equations. It has got to be of the correct form in order to take the inverse LT

It really depends a lot on the addressees of your lecture. For physicists, I'd introduce Fourier and Laplace transformations only, if they are advanced enough to have complex-function theory (particularly having the residue theorem at hand) and also some knowledge about generalized functions/distributions like the Dirac-$\delta$ function. If you'd have to cover these subjects, it may be too much time used for treating these foundations rather than differential equations.

The most neglected topic in the math education of physicists in my opinion is the theory of Lie groups and algebras. Such an ODE lecture would be the perfect place to introduce it, maybe using the special case of ODEs which can be derived from variational principles (most importantly the Hamilton principle of stationary action), where with the Poisson-bracket formalism of the Hamiltonian canonical equations you have the perfect setup for Lie-algebra and -group theory and Noether's theorems!

 

[samwinchester]

A first course in differential equations should cover the fundamental concepts and techniques necessary for solving ordinary differential equations (ODEs). Here are some key topics that could be included in such a course:

1. Introduction to ODEs: Definition, types of ODEs, order and degree of ODEs, initial and boundary value problems, and examples.
2. First-order ODEs: Separable equations, linear equations, homogeneous equations, exact equations, and Bernoulli equations. Applications in growth and decay problems, population models, and mixing problems.
3. Second-order ODEs: Homogeneous and nonhomogeneous equations, linear equations with constant coefficients, and applications in oscillation and mechanical systems.
4. Higher-order ODEs: Linear equations with constant coefficients, homogeneous and nonhomogeneous equations, and applications in electrical circuits and engineering.
5. Laplace Transform: Definition, properties, inverse transform, and applications in solving linear ODEs with constant coefficients and discontinuous forcing functions.
6. Series Solutions: Power series solutions, radius of convergence, and applications in solving ODEs with variable coefficients.
7. Numerical Methods: Euler's method, Runge-Kutta methods, and applications in solving ODEs numerically.

To motivate students, it's important to emphasize the real-world applications of differential equations in physics, engineering, biology, and other fields. Show them examples of how differential equations can model the behavior of physical systems and how the solutions can be used to make predictions and design solutions. It's also helpful to provide visualizations, such as phase diagrams and graphs, to illustrate the behavior of solutions.

To make the course well-structured and practical, consider organizing the topics around the types of ODEs and the techniques for solving them. Start with first-order equations and move to higher-order equations and more advanced techniques such as the Laplace transform and series solutions. Include plenty of examples and exercises to reinforce the concepts and develop problem-solving skills.

Overall, the goal should be to provide students with a solid foundation in differential equations and prepare them for more advanced courses in applied mathematics and physics.

 

[MidgetDwarf]

A readable book is also the one by Ross: Ordinary Differential Equations. It's only downside would maybe be the section dealing with Laplace Transform and Inverse Laplace. It's readable, but sadly many students do not know how to read a math book at that stage. The problem sets may be too easy, but it does a good job explaining the why. Particularly the section of solving non homogenous equations. The book is also cheap.

It also has material for a second course. Worth looking at, even as a reference.

Simons book is also a nice treat, but maybe a tad too difficult for todays students. Ross is a bit more standard

 

[Grelbr42]

Re: Laplace transform. Yes it's limited. As opposed to your favorite method that solves every possible DE? Even numerical methods only work for particular categories of DE.

The big deal about Laplace is that it gives you a closed form with explicit boundary condition dependence. If you recognize you have a situation that Laplace works on, it's celebration time. Usually you can tell if it's such a case fairly quickly, so you don't waste a lot of time.

 

[ohwilleke]

One relevant factor is the make up of the student body in the class.

For example, an Abstract Algebra class aimed at future high school mathematics teachers is going to have a different emphasis from one aimed at future physicists which would be different again from one targeted at future mathematicians who are primarily concerned with theory rather than physical applications.

The same is true of differential equations. Is this likely to be a terminal course for most of your students or are they likely to have further academic work that builds on it? Are they mostly engineers? Economists or biologists? Are there lots of ROTC students? High school math education majors? Is there anything else distinctive about this particular class of students?

 

[Will Flannery]

I'll give a controversial opinion .... assuming that the DE module is for a physics program. I'd begin by pointing out that classical physics is based on analyzing differential equation models of physical systems, but that, unfortunately, differential equation models of most physical systems are analytically unsolvable. I'd then point out that computers revolutionized physics outside the classroom 60 years ago by making it easy to analyze DE models of physical systems, even unsolvable ones.

Given that, as I see it you have 2 options - the first is to teach the traditional course in DEs.

The second is to teach a quickie course in ODE's, followed by the basic but powerful computational methods for solving ODEs and PDEs, both of which are very easy and intuitively transparent.

Taking the second option, I have written up what I think are the essential topics for a 1st course in calculus, starting from zero and going thru linear 2nd order systems, and it's only 17 pages long - https://www.academia.edu/94107426/Essential_Calculus_a_Revolutionary_Approach_to_Teaching_Calculus

 

[vanhees71]

I think, it's still important to have a good knowledge about the analytical properties of solutions of both ODEs and PDEs in physics. One should also be able to critically judge about the validity of numerical calculations. Of course it's today utmost important to also have good courses on numerics and programming, but these should be additional lectures and not the primary content of an introductory lecture on differential equations.

 

[Muu9]

The most neglected topic in the math education of physicists in my opinion is the theory of Lie groups and algebras. Such an ODE lecture would be the perfect place to introduce it, maybe using the special case of ODEs which can be derived from variational principles (most importantly the Hamilton principle of stationary action), where with the Poisson-bracket formalism of the Hamiltonian canonical equations you have the perfect setup for Lie-algebra and -group theory and Noether's theorems

This might be fine for the physics students who've taken classical mechanics, but what about the math majors who haven't?

 

[vanhees71]

I'd say for math majors group theory should be a standard topic anyway, and Lie groups are such a classic topic, combining the analysis on differentiable manifold with modern algebra, that also for pure mathematicians it should be an interesting topic.

 

[MidgetDwarf]

I'd say for math majors group theory should be a standard topic anyway, and Lie groups are such a classic topic, combining the analysis on differentiable manifold with modern algebra, that also for pure mathematicians it should be an interesting topic.

True, in regards to group theory being a standard topic. But the majority of math majors in the states, take intro ODE way before taking algebra. In regards to manifolds, many US universities no longer have a course in multivariable analysis, in which manifolds at minimum in R^n, are briefly talked about towards the end. Ie., generalized Stoke's Theorem.

But yes, these topics you mentioned are interesting.

 

[mpresic3]

A readable book is also the one by Ross: Ordinary Differential Equations. It's only downside would maybe be the section dealing with Laplace Transform and Inverse Laplace. It's readable, but sadly many students do not know how to read a math book at that stage. The problem sets may be too easy, but it does a good job explaining the why. Particularly the section of solving non homogenous equations. The book is also cheap.

It also has material for a second course. Worth looking at, even as a reference.

Simons book is also a nice treat, but maybe a tad too difficult for todays students. Ross is a bit more standard

Most books by Sheldon Ross are good. I am unfamiliar with this one. Do you mean Simmons or Simon's
Simmons book is good. I like that he included historical notes