22 votes

How would you explain what a PDE is to a very educated layman with no math background?

I would say something like this: "Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are ...
mweiss's user avatar
  • 17.4k
14 votes

Should I teach Laplace Transforms? How much?

Consider something besides an "all or nothing" approach. Here's what I did a couple of times when the topic was optional and I didn't have much time, but I still wanted to give students an ...
Dave L Renfro's user avatar
13 votes
Accepted

Diagram of Methods to Solve Differential Equations

I would just like to mention that other similar flowcharts have been developed, of varying degrees of generality, which you might consult. Here is one (by Adam Monahan). And another (by Jeremy ...
Joseph O'Rourke's user avatar
10 votes

Why do we study ordinary differential equations?

Of course I agree that one motivation for studying ODEs is that they have applications. But it might be useful to also point out another fact that students do not always think about: ODEs are often ...
Gustav's user avatar
  • 201
10 votes

Why do we study ordinary differential equations?

ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable). […] Am I missing another application […]? This may be somewhat pedantic, but I ...
Wrzlprmft's user avatar
  • 2,548
9 votes
Accepted

Why aren't integral equations often taught "back to back" with differential equations?

While there is a logical symmetry between differential equations and integral equations, it seems that (as they say) "laws of nature" are written in differential equations, not integral equations. As ...
paul garrett's user avatar
  • 14.6k
9 votes

What mathematical topics are important for succeeding in an undergrad PDE course?

They must understand the quadratic equation and how to factor it, solve it and manipulate it formally in both the real and complex case. Also, how to solve simple trigonometric equation and knowledge ...
James S. Cook's user avatar
8 votes
Accepted

How much prior math should I review in teaching a graduate-level course?

Even for math grad students, I'd forcefully review much more than many traditions seem to indicate. That is, I would not presume perfect recall of the standard curriculum, especially either in detail ...
paul garrett's user avatar
  • 14.6k
8 votes

Should I teach Laplace Transforms? How much?

Also having a short period of time to introduce my DE students to Laplace Transforms I began with two 'Axioms': (1) $\mathcal{L}\{c_1y_1(t)+c_2y_2(t)\}=c_1Y_1(s)+c_2Y_2(s)$ (2) $\mathcal{L}\{y^\...
John Wayland Bales's user avatar
8 votes

Why do we study ordinary differential equations?

Echoing many of the points made in the other excellent answers... but just to focus on one (to me very significant) aspect: differential equations characterize functions in (physically?) operationally ...
paul garrett's user avatar
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8 votes
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Is the Wronskian still assumed for graduate education?

I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true. Certainly, anyone who has studied a semester of ...
Alexander Woo's user avatar
7 votes

Why do we study ordinary differential equations?

Models, as you mention, are a huge source of applications. One can mention some in physics (free fall, radioactivity, pendulum, ...), in biology (cell growth), in chemistry (kinetics) among other. The ...
Benoît Kloeckner's user avatar
7 votes

Take-Home Examination on Ordinary Differential Equations?

Comment-answer, but too long for a comment: I think you are thinking about this wrong. Tests are some of the MOST valuable hours in a course. They are high stakes performances (like in music or ...
guest's user avatar
  • 139
6 votes

A Plan for a Treatise Study of the Classical Theory of PDEs

Despite the reputation that mathematics has for "logical order" and such, that assertion is misleading about how to study it, and how to understand it, I think. For one thing, it is not the case that ...
paul garrett's user avatar
  • 14.6k
6 votes

Why do we study ordinary differential equations?

Mention Newton's Second Law. $F = ma$ is an ODE (a trivial one if $F$ is constant, but nontrivial when $F$ or $m$ depend on position or velocity). Since one of the fundamental laws of nature is a ...
John Coleman's user avatar
  • 1,536
6 votes

Differential equations - definitions

One reason that calculus solution concepts didn't necessarily have labels of this kind is that such things are fairly ad-hoc. But all of the terms you listed - "linear", "homogeneous", etc. - have ...
kcrisman's user avatar
  • 5,976
6 votes

Online homework systems for ordinary differential equations

MyOpenMath is free. It doesn't look like it has a full course, but it has something, and you could add to it. Here's what I found: Covers chapters 1-6 and 8-10 from the Trench text (Elementary ...
Sue VanHattum's user avatar
  • 20.3k
5 votes

Differential equations - definitions

I'll give you the big picture of the differential equations course I've taught for a few years, I'm not sure what the curriculum is at your university, but, odds are we have much in common. First, ...
James S. Cook's user avatar
5 votes

Why do we study ordinary differential equations?

We should study Ordinary Differential Equations because it is beautiful mathematics which clearly illustrates the wondrous connection between analysis and algebra. Linear algebra, or perhaps matrix ...
James S. Cook's user avatar
5 votes

A good way to learn differential equations rigorously

A classic book is Coddington & Levinson, Theory of Ordinary Differential Equations. Probably tough going for most students. Coddington, An Introduction to Ordinary Differential Equations. Much ...
Gerald Edgar's user avatar
  • 7,587
5 votes

What mathematical topics are important for succeeding in an undergrad PDE course?

I would say that a good knowledge of linear algebra is paramount (for many reasons). About equally important is the full mastery of the differential and integral calculus of one variable. With these ...
fedja's user avatar
  • 3,869
4 votes

In what order should I teach methods for solving (linear) ODEs?

You left out linear first order ODE's, with their integrating factors. I use this often, compared to the other techniques. This probably should be done just after separation of variables, though I ...
Rory Daulton's user avatar
  • 2,582
4 votes

What is the right way to order the topics in a first ODEs course?

Your desire for some beautiful integrated thematic math person version of course shows not thinking about two important things: Your students and what they need. How people learn best. For number 1, ...
person's user avatar
  • 49
4 votes

Why do we study ordinary differential equations?

My slant is more tactical than the general idea of modeling. An ODE course is useful for students in engineering and physics because they will need the concepts to follow derivations and work ...
guest's user avatar
  • 284
4 votes

How would you explain what a PDE is to a very educated layman with no math background?

Imagine a surface, for example, a horse saddle. Note that the "curvature" of the surface differs not only when you measure it at different points, but also when you measure it along different ...
JRN's user avatar
  • 10.8k
4 votes
Accepted

Refreshing math knowledge

Don't underestimate how much your knowledge and skills can fade when unused, even after a couple of years, let alone twenty. I've experienced this personally when trying to study advanced topics for ...
J W's user avatar
  • 4,663
4 votes

Why are "homogeneous differential equations" in the standard ODE curriculum?

Introduction This is some sort of “apology” for teaching homogeneous ODEs. I think there’s a certain beauty and simplicity to them. That beauty is overlooked in most textbooks, which Rota sourly ...
user1815's user avatar
  • 5,545
4 votes
Accepted

How can I visualize differential equations and Integration in real life?

You have asked two very different questions. I'll leave the differential equations for someone else. There is one particular application of integration which is my favorite last problem to do in Calc ...
Sue VanHattum's user avatar
  • 20.3k

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