It’s hard to be believe we haven’t already done this one, since it’s been irritating us for years. But in any case it really irritated us again yesterday during a tute, and so here we are.
The following introduction and example, chosen largely but not entirely at random, is from Cambridge’s Specialist Mathematics 3 & 4.
Commenters have pointed out a number of irritants and more serious issues, and there are others, which we’ll get to in Part 3. The line by line issues are secondary, however. The primary problem, as commenter anonymouskouri was getting at, is that the entire approach is fundamentally screwy. This screwiness turns out to be way more than an issue of mathematical integrity. It actively confuses the hell out of students.1
First, some scene-setting background. VCE Specialist Mathematics includes the study of Differential Equations, but such a title is way overselling the breadth of the topic. Excepting for those to be solved by direct antidifferentiation the only DEs that are required to be solved are separable, that is are of the form,
Yes, there’s Euler’s method and slope fields and some simple modelling. But in terms of solving DEs, separable is all there is, and they are easy. Allowing ourselves a little black magic for the moment, we cross multiply to get the x‘s on one side and the y‘s on the other:
Then, we integrate:
That’s it. Everything else, the actual integration and solving for y and so forth, is simply detail.
Now, before getting on with hammering Cambridge, let’s demystify the black magic at least a little. Of course students are taught to pretend to never to think of dy/dx as a fraction, and they would/should never write the middle line (2): they only think the middle line (without telling anyone), and then they write the integral equation (3). But (3) is easy to justify.
Starting with (1), we can divide both sides by g(y) and integrate with respect to x. That gives,
But now the LHS of (2′) is exactly set up for integration by substitution, with the (as yet unknown) function y being the new “variable”. (Think of the substitution being u = y, if that helps clarify what’s going on.) The result of applying substitution to the left side of (2′) gives us exactly (3).2
Now, one may argue that we have simply replaced the black magic of separation with the black magic of substitution. That’s not really the case: the only step in going from (2′) to (3) is exactly applying integration by substitution (albeit not in the function notation that makes the legitimacy clearer). Of course trying to think why substitution works this way is not at all obvious.3 But in any case, the magic of separation is, at worst, no blacker than the magic of substitution. And the solving of DE’s in Specialist Mathematics contains nothing else of particular note.
1. This WitCH was motivated generally by seeing years of confused students, and it was specifically motivated by a tutorial the previous day. That tutorial was for a very good student, who is at a top notch school, who has a top notch Specialist teacher, and nonetheless the student had essentially no clue what was going on with differential equations.
2. Cambridge later provides this computation, using (2′) to justify (3), but they do so without even a hint that integration by substitution is the key ingredient. Which is stupid. The entire Cambridge chapter on DEs is a WitCH-infested mess.
3. In the comments we’ve promised to do a post on the black magic of integration by substitution. We’ll get to that as soon as we can.
Now, to the poisonous heart of the WitCH. Cambridge begins the DEs chapter with “verifying” solutions,4 after which they take care of the trivial cases, y’ = f(x) and y” = f(x). Then, it’s on to separable DEs.
As indicated by the excerpt above, Cambridge begins by considering DEs of the form
Of course, (1′) is a special case of (1), so its solution and the method of obtaining it are also special, but not notably simpler, cases:
As such, there is no particular need to introduce (1′) as a first and special type. Nonetheless, it’s a standard and reasonable enough approach. What is not remotely reasonable, however, is to consider (1′) by also introducing a gratuitously different, and woefully unexplained, method of solution. There is simply no need for the method Cambridge employs and it is actively, hugely confusing to do so.
What Cambridge is applying is the inverse function theorem, but without a hint that this is what they are doing,5 or explaining at all what follows. It is all utterly pointless, since substitution is conceptually easier, and is required anyway to solve the general type, (1). Just to be absolutely clear, Cambridge‘s method of solving (1’) is legal, or at least legalisable. What it is not, is sane.
The absurd cherry on top is that when Cambridge finally gets around to general separable DEs they do not even hint that (1′) is separable, that it is a special case of (1). With the exception of the quick and inevitably ignored note in the excerpt above, (1′) is considered to be an entirely different form of DE, up to and including in the chapter summary:
An entire goddam page when one equation would suffice. Pure madness.
4. As VCE teachers and students are all too aware, one has to pay careful attention to VCAA’s prissy and perverted language. In particular, in VCE “verify” does not (usually) mean the same as “show that”; see here, for one of a thousand examples. This language perversion also screws up kids doing DEs, who commonly enough over-cautiously interpret “verify” to mean “solve”.
5. The inverse function theorem is never mentioned in VCE. It is always simply assumed that inverse functions can be differentiated by means of the chain rule.
To finish off, we’ll list the many line by line irritations. Thanks again to the commenters who worked hard to make sense of this nonsense.
(a) “Identity” is a very weird name for the formula for the derivative of an inverse.
(b) Cambridge‘s use of pronouns sucks: “this becomes” should be “the differential equation becomes”.
(c) No explanation is given for why dx/dy = 1/g leads to x = ∫1/g. Yes, it’s just definition, but in the context this will be far from obvious to most VCE students and teachers.
(d) “Differential equations can be constructed from statements” is one of the all-time great stupid sentences.
(e) As indicated, the example begins Cambridge‘s discussion of modelling. Why on Earth would one begin modelling with an entirely unmotivated and thoroughly implausible population model? Why would one begin with a “statement”, rather than an explicitly expressed model, with the parameter explicitly given?
(f) Even given the set up, the use of the “proportional to” symbol is pointless and distracting.
(g) “population-time graph” needs an n-dash, not an m-dash.
(h) In VCE it is standard for “increasing” to not mean strictly increasing; thus, the conclusion that k > 0 is questionable. There is also no particular benefit or extra naturalness in assuming the population is increasing. To do so, and thus to have to fuss about k > 0, is just muddying the message of this very first modelling example.
(i) Cambridge‘s use of pronouns still sucks: “Note that it is of the form” should be “Note that the differential equation is of the form”.
(j) Flipping the derivative has encouraged the needless and annoying fiddling with a 1/k.
(k) “Therefore” dots suck.
(l) Flipping the derivative has effectively turned the +c into a -c. These things matter. (In a moment c is gonna be negative.)
(m) It is not pointed out that, since t ≥ 0, we also require c ≤ 0.
(n) “Therefore” dots still suck.
(o) The graph is wrong, falsely suggesting that the turning point should be on the P axis.
(p) No consideration is given for the possibility of the solution P = 0, or the means of obtaining it, or the fact that this solution is not contained in the “general” solution given, or that the initial condition P(0) = 0 leads to two entirely different solutions, which kinda sucks for a model. One might wish to rule all this out with the “increasing” business, but it still has to be explicit and argued.