21
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 ...
- 17.2k
16
votes
Commonly taught method divides by zero
The method is mathematically incorrect, but whether it is wrong or not, depends on whether people know what they're doing or not.
The possible division by zero is a mistake, yes. But it is a mistake ...
- 824
13
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 ...
- 5,698
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 ...
- 29k
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 ...
- 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 ...
- 2,538
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 ...
- 14k
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 ...
- 10.7k
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^\...
8
votes
Accepted
Second Order Differential Equation Example Request
Some examples from Erwin Kreyszig's Advanced Engineering Mathematics 7th ed. (John Wiley & Sons, Inc., 1993):
Falling stone
$\frac{\mathrm{d}^2y}{\mathrm{d}t^2}=g$
where $y$ is the position of ...
- 10.7k
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 ...
- 14k
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 ...
- 14k
8
votes
Accepted
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 ...
- 2,474
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 ...
- 8,989
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 ...
- 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 ...
- 14k
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 ...
- 5,952
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 ...
- 1,506
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 ...
- 19.4k
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 ...
- 10.7k
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, ...
- 10.7k
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 ...
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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 ...
- 3,439
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 ...
- 2,552
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, ...
- 49
4
votes
Theme of a Differential Equations Course
These are some of the questions I would use to guide the course:
What is a differential equation ? In particular, what language should we use to communicate the structure of a differential equation.
...
- 10.7k
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 ...
- 10.7k
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