Why DEs are Important: Realigning the Focus of University Teaching

[Unstable]

After a short look on the question formulated in this differential equation forum I came to the conclusion that the teaching of DEs at Universities goes in a wrong direction. The aim of DEs is not “how could I solve them” the point is “what I could do with them”. In practice nobody solves DEs analytically (except of simple linear systems) and it is actually not necessary to know all the tricks and techniques. Students should learn what the DEs describe and how they could be applied to perform modeling of certain phenomena. And in an advanced case, how could I build my own DEs. Just my two cents.

 

[MarneMath]

Maybe my Differential Equations class was different, but our focus was one proving existence, uniquness, reduction of order, finding maximum and minimal, etc. There existed a few problems that were about finding analytical answers, but the majority of it was based on proving theorems within differential equations.

Perhaps this type of class is not needed for a physics/engineer major, but it made sense for a math major. I've also found that physics departments offer classes that teach numerical methods and the more physical aspects of DE.

 

[Mute]

Why is the quadratic formula necessary? In practice nobody can solve for the roots of higher order polynomials, so why both learning the quadratic formula or even polynomial division if we're just going to have to use a computer to solve them anyways?

For a more serious answer to the question, the DEs that do have analytical solutions are often very important, and the ones where we need to use methods like power series solution and the Frobenius method also tend to be rather important, and it's important to know where these solutions came from. Knowing various tricks and techniques also enables you to show that some DEs are just a different from of a DE you already know, and that's pretty important.

While most DEs cannot be solved analytically, many are still related to or close to solvable linear DEs in some way, which means tools like perturbation theory or asymptotic analysis can build on these results. This is rather important, one of the reasons being that understanding the cases that you can treat analytically, even only in an approximation scheme, are very important for verifying that your method of numerically solving a DE that you can't treat analytically is actually behaving correctly and not giving something that looks reasonable but is actually nonsense.

 

[arildno]

Why should you bother about grammar and syntax?
Can't you just write the Great Novel of the Century?

Learning basic structures and solution techniques of DEs builds up your skills to implement them fruitfully, and independently, in later work.
Numerical methods are NOT trivial stuff (not the least, questions of stability, well-posedness and so on), and much of the actual justification of such methods relies intimately on knowledge gained by the study of simpler DEs.

 

[Unstable]

Thanks so far the answers. I do not totally disagree with the presented arguments but I still have the opinion that this important field in mathematics, which has so much great potential in application, is educated in a wrong way at Universities.

I am a mathematician who learned the differential equations in the classical way, after two weeks of the first course I knew how to proof Picard-Lindelöf and so on... bla bla.. the typical theoretical stuff. Also we had 2-3 examples like Lotka-Volterra … .After that I learned things about Ljapunov Stability and finally I had courses about numerical methods for different types of DEs.

However, the first time I was introduced to really apply DEs e.g. for modeling and simulation was during my PhD in math. Here we started to build our own systems based on physical laws or biological imaginations. This was the first time were I saw would you could actually do with these things.

What I want to say is that the modeling/simulation approach, at least in my education, was completely ignored. This is actually a shame because this is what the industry wants. Professors always wonder why so many talented mathematicians end up in consulting companies together with lawyers etc. . The answer is so obvious because their education is wrong. Nearly most of the jobs where knowledge about DEs is necessary are occupied by physicians, engineers, …

I remember a talk some years ago from a director from a Fraunhofer Institut presenting to Professors from Mathematics departments. One sentence was something like “Stop educated your students as they will become all professors.” And there is so much truth in this! From my point of view the study of mathematics is in most universities (not in all, there are several departments with great new concepts) to classical and therefore, to antiquated.

 

[Runei]

On Control Systems engineering the first thing you learn is how to model electrical, mechanical, thermal and pneumatic systems. So I quickly found out what they're used for.

Another thing: If you had to implement the computer programs that solve the DE's you would have to know how to. Computers don't just do it, they have to solve by the numerical methods - so its important that designers of these computer programs know how it actually works :)

 

[Studiot]

My thoughts on this issue are from a slightly different perspective from any so far offered.

Firstly any DE can also be presented as an integral equation. Sometimes these are easier to handle (note I didn't say solve).

When translated to the numerical/ computer world we have a FE methods and BE methods.
These are analogous to Greens/Gauss/Stokes theorems connecting effects on the boundary with internal effects.

In Sructural Engineering the decision to build a reinforced concrete frame or a steel frame is subject to fashion. There used to be two industry trade associations which sponsored educational material in concrete or steel structures and provided design tables etc. When one was ascendent most buildings had either a steel or concrete structure.

I think that there is a huge weight of educational and other technical material available that pushes the teaching and use of this subject in a certain direction.
It is, after all, easier to draw on existing than to draw up new.

So there there is much more material and software about DEs compared with Integral Equations. There is considerable more material about FE than BE methods.

 

[arildno]

Thanks so far the answers. I do not totally disagree with the presented arguments but I still have the opinion that this important field in mathematics, which has so much great potential in application, is educated in a wrong way at Universities.

I am a mathematician who learned the differential equations in the classical way, after two weeks of the first course I knew how to proof Picard-Lindelöf and so on... bla bla.. the typical theoretical stuff. Also we had 2-3 examples like Lotka-Volterra … .After that I learned things about Ljapunov Stability and finally I had courses about numerical methods for different types of DEs.

However, the first time I was introduced to really apply DEs e.g. for modeling and simulation was during my PhD in math. Here we started to build our own systems based on physical laws or biological imaginations. This was the first time were I saw would you could actually do with these things.

What I want to say is that the modeling/simulation approach, at least in my education, was completely ignored. This is actually a shame because this is what the industry wants. Professors always wonder why so many talented mathematicians end up in consulting companies together with lawyers etc. . The answer is so obvious because their education is wrong. Nearly most of the jobs where knowledge about DEs is necessary are occupied by physicians, engineers, …

I remember a talk some years ago from a director from a Fraunhofer Institut presenting to Professors from Mathematics departments. One sentence was something like “Stop educated your students as they will become all professors.” And there is so much truth in this! From my point of view the study of mathematics is in most universities (not in all, there are several departments with great new concepts) to classical and therefore, to antiquated.

You wouldn't have been able to learn the applications, and the developed your own fruitful approaches WITHOUT the theoretical courses.

 

[Unstable]

You wouldn't have been able to learn the applications, and the developed your own fruitful approaches WITHOUT the theoretical courses.

I know that this is the opinion of many people but it is definitely NOT TRUE!

Don't get me wrong, I just want to say that in DE related subjects in universities not enough
focus is given to application which is a tremendous disadvantage expecially for mathematicans.