Basic Skills in Mathematics, Problems 23–24 (Part II)
In the previous post we obtained the solution to Problem 23. I originally intended to go on and solve Problem 24 in a similar way, but looking at the answer to Problem 23, I felt I'd spotted something new, so I held off. Today, after spending a class period thinking it over, I arrived at some new results about this type of quasi-homogeneous differential equation, so I'm opening a new post to share them with you.
I. The General Solution of a Special Class of Quasi-Homogeneous Differential Equations
In the previous post, we found the solution to the quasi-homogeneous differential equation $\frac{dy}{dx}=x+\frac{x^3}{y}$:
$$(2y+x^2)(x^2-y)^2=C$$
or, written in this form:
$$(y+\frac{1}{2} x^2)(y-x^2)^2=C$$
where $y=-\frac{1}{2} x^2$ and $y=x^2$ are two particular solutions of the differential equation. This general solution has an obvious pattern, which led me to conjecture that the differential equation
$$\frac{dy}{dx}=Ax^m+\frac{B x^{2m+1}}{y}$$
has two particular solutions, $y=c_1 x^{m+1}$ and $y=c_2 x^{m+1}$, and that its general solution can be written as
$$(y-c_1 x^{m+1})^{\alpha} (y-c_2 x^{m+1})^{\beta}=C$$
where $c_1,c_2$ are the two roots of the algebraic equation $(m+1)c^2=Ac+B$, and $\alpha,\beta$ is an undetermined constant.
To check whether this conjecture is correct, rather than solving the differential equation from scratch, we instead directly transform the conjectured solution and see whether it satisfies the original equation. Differentiating both sides of the conjectured solution gives:
$$\begin{eqnarray*}\left\{ {\alpha \left[ {\frac{{dy}}{{dx}} - {c_1}(m + 1){x^m}} \right](y - {c_2}{x^{m + 1}}) + \beta \left[ {\frac{{dy}}{{dx}} - {c_2}(m + 1){x^m}} \right](y - {c_1}{x^{m + 1}})} \right\}\\ \times (y - c_1 x^{m + 1})^{\alpha - 1}(y - c_2 x^{m + 1})^{\beta - 1}=0\end{eqnarray*}$$
Without loss of generality, we should have
$$ {\alpha \left[ {\frac{{dy}}{{dx}} - {c_1}(m + 1){x^m}} \right](y - {c_2}{x^{m + 1}}) + \beta \left[ {\frac{{dy}}{{dx}} - {c_2}(m + 1){x^m}} \right](y - {c_1}{x^{m + 1}})} =0$$
which rearranges to
$$\begin{eqnarray*}(\alpha+\beta)y\frac{dy}{dx}-(\alpha c_1 +\beta c_2)(m+1) x^m y-(\alpha c_2 +\beta c_1)x^{m+1} \frac{dy}{dx}\\ +(\alpha+\beta)c_1 c_2 (m+1) x^{2m+1}=0\end{eqnarray*}$$
Comparing with the original differential equation $y\frac{dy}{dx}=Ax^m y+B x^{2m+1}$, we should expect:
$$\frac{(\alpha c_1 +\beta c_2)(m+1)}{\alpha+\beta}=A,\alpha c_2 +\beta c_1=0,c_1 c_2 (m+1)=-B$$
Since $c_1,c_2$ are the two roots of $(m+1)c^2=Ac+B$, the third equation is automatically satisfied. We also have $A=(c_1 +c_2)(m+1)$, and substituting this into the first equation and simplifying gives $\alpha c_2 +\beta c_1=0$, which is exactly the second equation, so the three equations are mutually consistent. All we need is to find one solution of $\alpha,\beta$; from $\alpha c_2 +\beta c_1=0$ we can obtain $\alpha=k c_1 ,\beta=-k c_2$, and hence the general solution of the original equation is:
$$(y-c_1 x^{m+1})^{k c_1} (y-c_2 x^{m+1})^{-k c_2}=C$$
where k is a nonzero constant chosen to make the form of $k c_1,k c_2$ simpler.
II. A Few Random Musings
This post originally grew out of Problems 23 and 24 in 100 Basic Skills Problems in Mathematics. Thanks to a certain mathematical sensitivity and persistence, I ended up with this fairly elegant final result. From the initial particular solutions, to yesterday's derivation of the general solution via a change of variables, to today's conjecture and proof of the general pattern — the whole process was full of excitement, and perhaps it's precisely this kind of excitement that keeps me tirelessly pursuing mathematics and physics.
While I'm at it, let me vent a bit about the whole business of taking notes in class. There's a saying in universities: "if you don't know how to skip class, you don't know how to attend class." And yet I've basically never skipped a single class, whether it was a general-education course or one in my major. At the same time, though, I've basically never listened attentively to any class either, especially not the courses in my major. The reason I go to class at all is that it lets me avoid the various hassles that come with being marked absent, while also giving me time to sit quietly and think through my own problems — why wouldn't I take that deal? So there's simply no way I'm going to take notes. In fact, after all these years of schooling, I've almost never taken a single note. In today's Advanced Algebra class, the instructor criticized our class for its bad habit of not taking notes. I don't know what everyone else thinks, but I don't entirely agree with him. Taking notes is neither necessary nor sufficient. My not taking notes isn't because I consider myself some kind of genius — quite the opposite, I consider myself perfectly ordinary. And precisely because I'm ordinary, if I tried to listen attentively and take careful notes, there'd be no way I could keep up with the instructor's train of thought. Or rather, even if I managed to keep up, I still wouldn't have formed my own way of thinking. And anything like that never really belongs to me.
I still very much agree with what Feynman said: "What I cannot create, I do not understand." Only what you create yourself truly belongs to you. Creation means your own mind has personally gone through the entire process of reasoning — understanding "what this is," "why it can be done this way," "why it can't be done that way," "how far a mistaken approach can be pushed before it fails," and so on. So I can only put my head down and think slowly. My pace is slow, and sometimes I fall quite far behind, but when I do manage to arrive at an idea, I've solved a great many problems all at once. The crucial thing is that the idea is mine — that way of thinking belongs to me. It's not that what the instructor teaches isn't good; it's that I'm too mediocre to keep pace with the instructor.
However much things vary, they all trace back to the same root. Take geometry, for instance: in principle, Euclidean geometry rests on just five axioms, and starting from these axioms, with a certain amount of deductive reasoning, one can arrive at every result in Euclidean geometry. Of course, we don't approach every problem starting from the very bottom — we also learn all sorts of corollaries and theorems that help us along the way. But what matters most is still the thinking, the reasoning itself, which no language can fully capture — you can only come to understand it by experiencing it yourself. So what gets written down in notes is still just "the fish"; the real "fishing," the skill itself, lies in deep thought. And there's something a bit dangerous here too: many people feel reassured once they've taken notes, but in reality they never actually understand what they've written down — they only pull the notes out and flip through them when the exam is approaching. Notes used that way have lost their original purpose entirely.
Notes are meant to be a temporary memory aid; once you've truly understood something, once you've reached the point where you have created it, the notes can be thrown away. I may not always manage to live up to everything I've said here, but I'll do my best.
People are always saying a good memory is no match for a poor pen. But let me ask you this: do you want a good memory, or do you want a poor pen?
Just some random thoughts — criticism welcome.
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.