# Tag Info

21

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 varying. For example the rate at which a drug metabolizes may depend not only on how much of the drug is in the body but also on how much blood sugar is in the body,...

16

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 one can ignore if in the end you get a solution to the differential equation. To make it full proof, what I do is the following sequence of steps. Possibly ...

14

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 Higgins): And another (by Enrique Areyan):

13

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 introduction to the method. Simply restrict yourself to introducing the method by defining the Laplace transform, computing it for some simple examples, explaining what ...

10

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 think that your answer is so generic that it covers every possible application per construction: The fundamental formulation of an ODE has the change of the ...

9

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 the most succinct way of defining functions. You could use the exponential function as an example, because your students will have seen it already. Ask them to ...

9

Benjamin Dickman already linked to my answer on Math.SE, but I didn't talk too much about exactly the motivation of the characterisation for second order equations. So let me write a little bit about that here. First, I want to emphasize again that the classification is incomplete: the usual scheme does not include all possible equations. Let us start by ...

9

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 far as I can tell, rigorous study of integral equations arose in the late 1800's in part because of the relative difficulty of rigorous study of (especially ...

8

For a thoughtful view on the subject, see Gian-Carlo Rota, "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations." The tone of the linked text may be off-putting to some, but the ideas within are a good starting point. I share many of his frustrations and agree with many of his conclusions. I've pasted the "ten lessons" below; ...

8

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 the stone and $g$ is the gravitational acceleration Hanging cable $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=k\sqrt{1+\frac{\mathrm{d}y}{\mathrm{d}x}^2}$ where $y$ is ...

8

So, there is some point where they just propose multiplying the given solution by $t$ and then work out the generalized e-vector of order two condition. Pretty in its way, but I prefer the perspective of the matrix exponential. Define $e^{tA}=I+tA+(t^2/2)A^2+ \cdots$. It is easy to show that the matrix exponential is a solution matrix and it is an ...

8

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 or in "big picture". Further, in my experience, even very smart people with unusually good memories greatly benefit from repetition. It's not "one and done", ......

7

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 (as it turns out) meaningful ways. Other answers have noted the "dynamical" aspect, and @gustav noted in particular that some things are best described by ...

7

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) $\mathcal{L}\{y^\prime\}=sY(s)-y(0)$. From these two we derived the Laplace transforms for polynomial, exponential and sinusoidal functions and proceeded to use those results and ...

7

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 sports). Preparation for them drives a lot of learning. Then the actual execution and subsequent feedback is often much more valuable training than routine ...

6

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 beauty of the subject, tying together several branches of maths that student tend to consider separate, is of course also a strong motivation. Let me stress ...

6

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 there is a "true natural logical order", but only that things are order-able, in various ways, depending on tastes/prejudices. And, then, still, I doubt that ...

6

I agree that the standard textbook approach is not too exciting from a pure math perspective. I would say the order of the topics is largely based on an attempt to keep the subject from getting too abstract too soon. Also, I would wager the applications which are solved by the methods also benefit from the parsing of topics. For example, the first order ...

6

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 deep meaning in the algebra of solutions. (Indeed, given your current state of studies, you should be asking about the relationship between differential ...

5

This is not a complete answer, just a few pointers of what has worked for me. Different students react differently to different "methods" of solving linear equations. Should one learn to find the homogeneous solution first, or do the exponential-of-the-unmotivated-integral, or...? Other solvable families have the same problem. Moreover, I want to get to the ...

5

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 theory, when combined with calculus provides abstractions of ordinary functions which behave in ways similar yet fantastically different than ordinary functions. ...

5

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 differential equation, it makes sense that you should study differential equations if you want to understand nature. I don't know who said it first but I've always ...

5

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, our general goal is to solve differential equations. Depending on the problem, that may mean we seek a function whose derivatives satisfy the given DEqn, but, it ...

5

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 I. (We got behind this semester, and I was very sad not to have time for this. It feels like a perfect grand finale to me.) You probably learned the formula for ...

4

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 have seen it done before that topic. I see why you would want to delay this topic until after you study inhomogeneous linear ODE's, for consistency. But the ...

4

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. What is a solution to a differential equation ? In particular, what object is best suited to capture our intuitive concept of a solution? Can we free ourselves ...

4

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, they are mostly engineering and physics students. They benefit from exposure to the tricks of the trade so that when they have derivations and problems in ...

4

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 directions. For example, at the "center" of the saddle, the surface is curving upwards if measured along the "horse's spine" and the surface is curving downwards if ...

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