In the previous article, we obtained the solution to Problem 23. Originally, I intended to proceed similarly to solve Problem 24, but looking at the answer to Problem 23, I seemed to discover something new, so I did not continue writing immediately. After spending a class period today researching it, I obtained some new results regarding this type of quasi-homogeneous differential equation, and thus I am starting a new article to share them with everyone.
General Solution of a Special Quasi-Homogeneous Differential Equation
In the previous article, 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, so I conjectured 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 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 are undetermined constants.
To test the correctness of this conjecture, we will not solve the differential equation directly, but rather differentiate the conjectured solution to see if it satisfies the original differential equation. Differentiating both sides of the conjectured solution, we get: \begin{aligned} \left\{ \alpha \left[ \frac{dy}{dx} - c_1(m + 1)x^m \right](y - c_2x^{m + 1}) + \beta \left[ \frac{dy}{dx} - c_2(m + 1)x^m \right](y - c_1x^{m + 1}) \right\} \\ \times (y - c_1 x^{m + 1})^{\alpha - 1}(y - c_2 x^{m + 1})^{\beta - 1}=0 \end{aligned}
Without loss of generality, we should have: \alpha \left[ \frac{dy}{dx} - c_1(m + 1)x^m \right](y - c_2x^{m + 1}) + \beta \left[ \frac{dy}{dx} - c_2(m + 1)x^m \right](y - c_1x^{m + 1}) = 0
Rearranging this gives: \begin{aligned} (\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{aligned}
Comparing this with the original differential equation y\frac{dy}{dx}=Ax^m y+B x^{2m+1}, we expect: \frac{(\alpha c_1 +\beta c_2)(m+1)}{\alpha+\beta}=A, \quad \alpha c_2 +\beta c_1=0, \quad 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 obviously satisfied. At the same time, we have A=(c_1 + c_2)(m+1). Substituting this into the first equation and simplifying, we get \alpha c_2 + \beta c_1 = 0, which is exactly the second equation. Therefore, these three equations are self-consistent. We only need to find one set of solutions for \alpha, \beta. From \alpha c_2 + \beta c_1 = 0, we can obtain \alpha = k c_1, \beta = -k c_2. Thus, 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
Here, k is a non-zero constant chosen to make the forms of k c_1 and k c_2 simpler.
Random Reflections
The origin of this article lies in Problems 23 and 24 of the “100 Problems of Basic Mathematical Skills.” Due to mathematical sensitivity and persistence, I arrived at this rather elegant final result. From the initial particular solution to yesterday’s change of variables to find the general solution, and finally to today’s conjecture and proof of the general rule, the entire process has been filled with excitement. Perhaps it is this sense of excitement that keeps me tirelessly researching mathematics and physics.
On a side note, I’d like to comment on the idea of taking notes in class. There is a saying in university: “If you don’t know how to skip class, you don’t know how to attend class.” However, I have basically never skipped any classes, whether they were non-major or major courses. But at the same time, I have basically never paid close attention to any lectures, especially major courses. The reason I attend major courses is that I can avoid the various problems caused by absence while simultaneously focusing on thinking about my own problems in class; why not? Therefore, it is impossible for me to take any notes. In fact, in all my years of schooling, I have almost never taken notes. In today’s Higher Algebra class, the teacher criticized the bad habit of not taking notes in our class. I don’t know what others think, but I don’t quite agree with him. Taking notes is neither necessary nor sufficient. I don’t refrain from taking notes because I consider myself a genius; on the contrary, I consider myself ordinary. Precisely because I am ordinary, if I were to focus solely on listening and taking notes, it would be impossible to keep up with the teacher’s thinking; or rather, even if I kept up, I wouldn’t have formed my own way of thinking. Such things would never truly belong to me.
I still very much agree with Richard Feynman’s quote: “What I cannot create, I do not understand.” Only things I create myself can be considered mine. Creation means that my own mind has personally experienced the entire process of reasoning and thinking, understanding “what this is,” “why it can be this way,” “why it cannot be that way,” “how far a wrong approach can go,” and so on. Therefore, I can only lower my head and think slowly. My pace is slow, and sometimes I fall far behind, but when I successfully develop an idea, I have solved many problems. The key is that the idea belongs to me, and that way of thinking belongs to me. It’s not that what the teacher teaches is bad, but that I am too mediocre to keep up with the teacher.
Everything changes but the essence remains the same. For example, in geometry, in principle, Euclidean geometry has only five axioms. Starting from these axioms and adding certain deductive reasoning, one can obtain all the conclusions of Euclidean geometry. Of course, we don’t start every problem from the very bottom; we also learn various corollaries and theorems to assist us. But more importantly, there is thinking and reasoning, which is difficult to describe accurately in any language and can only be experienced personally. Therefore, what is recorded in notes is still just the “fish”; the true “fishing” is deep thinking. Furthermore, there is a dangerous aspect: many people feel at ease once they have taken notes, but in fact, they never understand what they have written until they pull them out to flip through them right before an exam. Such notes lose their original meaning.
Notes are for temporary reminders. Once you understand something and reach the point where “You have created it,” the notes can be thrown away. I may not be able to do everything I say, but I will try.
People always say “a good memory is not as good as a worn-out pen,” but I want to ask you: do you want a good memory or a worn-out pen?
Just some random thoughts; criticisms are welcome.
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