Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see without calculations that this equation is symmetric in $x$, $y$, or, say, that it is affine invariant? Quick calculation comes from considering Wronskian of 6 functions $1,x,y,x^2,y^2,xy$ which vanishes iff $(x,y)$ lie on a conic. But it is not quite satisfactory explanation.



differential equation of conics
I didn't see this question when it originally appeared, but an edit today brought it to the front page of "new questions" where I saw it. Back around 2005 or 2006 I came across this differential equation for conics (see this [26 April 2008 sci.math post](https://groups.google.com/g/sci.math/c/LwkcylTv_F8/m/66A8lWa1UfcJ), where I gave a formal-algebraic derivation of it), and ever since then I've been collecting copies of papers that discuss this equation when I happen to come across them. Although I have not made a systematic literature search on this topic, the references below (listed in chronological order) should give anyone interested in doing such a literature search a good starting point. In this regard, Mukhopadhyay **[5]** seems like the best place to begin.


**[1]** [Georges Henri Halphen (1844-1889)](https://web.archive.org/web/20180120232838/http://www-groups.dcs.st-and.ac.uk:80/history/Biographies/Halphen.html), [*Sur l'équation différentielle des coniques*](https://eudml.org/doc/85428) [On the differential equation of the conics], **Bulletin de la Société Mathématique de France** 7 (1879), 83-85. [MR1503806](https://mathscinet.ams.org/mathscinet-getitem?mr=1503806); [JFM 11.0250.02](https://zbmath.org/11.0250.02)



> 
> Halphen states that, in a paper in Volume 4 of BSMF (neither the paper's title nor a page number is given), he had indicated a way of obtaining a 5th degree differential equation whose solutions give all conics. The present paper gives another way to obtain the same equation. Beginning with $y = ax + b + \sqrt{Ax^2 + 2Bx + C},$ we differentiate twice to obtain $y'' = (AC – B^2)(Ax^2 + 2Bx + C)^{-\frac{3}{2}}.$ Thus, $(y'')^{-\frac{2}{3}}$ is an arbitrary 2nd degree polynomial, and hence we get $\left[(y'')^{-\frac{2}{3}}\right]''' = 0.$ When expanded, this becomes $9(y'')^2y''''' – 45y''y'''y'''' + 40(y''')^3 = 0.$ Halphen also remarks that $\left[(y'')^{-\frac{2}{3}}\right]'' = 0$ is a differential equation for all parabolas. The remaining 60% to 70% of Halpen's paper is devoted to showing how one can solve the differential equation for conics by certain substitutions and the use of standard methods.
> 
> 
> 


**[2]** [Thomas Muir (1844-1934)](https://en.wikipedia.org/wiki/Thomas_Muir_%28mathematician%29), [*The differential equation of a conic*](https://books.google.com/books?id=m6lJAAAAYAAJ&pg=PA143), **Philosophical Magazine** (5) 21 #129 (February 1886), 143-145. [JFM 18.0244.01](https://zbmath.org/18.0244.01)



> 
> **(first sentence of the paper)** THE differential equation of a conic,
> $$9\left(\frac{d^{2}y}{dx^2}\right)^{2}\frac{d^{5}y}{dx^5} \;\; - \;\; 45\frac{d^{2}y}{dx^2}\frac{d^{3}y}{dx^3}\frac{d^{4}y}{dx^4} \;\; + \;\; 40\left(\frac{d^{3}y}{dx^3}\right)^3 \;\; = \;\; 0,$$
> which Boole attributes to Monge, and to which Sylvester in this week's 'Nature' (vol. xxxiii. p. 224) so fully directs attention, can be readily obtained as follows.
> 
> 
> 



> 
> **(last paragraph of the paper)** In the 'Proceedings of the Edinb. Math. Soc.' 1884-85, pp. 95-100, Professor Chrystal has a note on "A Method for obtaining the Differential Equation of an Algebraical Curve." In the case of the cubic he gets for the non-zero side of the equation a determinant of the 6th order which he says can be simplified further. I find that when this simplification is made, and $y\_{2}, \, y\_{3}, \, \ldots$ are expressed in terms of $\alpha, \, \beta, \, \ldots,$ the result agrees perfectly with the above\*.
> 
> 
> 



> 
> **footnote \* states:** The differential equation of the cubic appears to have been first given by Mr. Samuel Roberts in the 'Educational Times.' The question and solution are reprinted in 'Math. from Educ. Times,' x. pp. 47, 48. The date of the first publication of the question, which unfortunately can in no case be got with certainty from the Reprint, should probably be 1868.---T. M. (January 18).
> 
> 
> 


**[3]** [Allan Joseph Champneys Cunningham (1842-1928)](https://en.wikipedia.org/wiki/Allan_Joseph_Champneys_Cunningham), [*Depression of differential equations*](https://books.google.com/books?id=lkQ3AQAAMAAJ&pg=PA118), **Messenger of Mathematics** 17 (1888), 118-145. [JFM 19.0296.01](https://zbmath.org/19.0296.01)


[another link to the paper](https://gdz.sub.uni-goettingen.de/id/PPN599484047_0017)



> 
> The differential equation of conics is dealt with on pp. 140-145.
> 
> 
> 



> 
> **(from p. 140)** As an example of these principles, the equation---
> $$9{y''}^{2}y^v \; - \; 45y''y'''y^{iv} \; + \; 40{y'''}^{3} \; = \; 0,$$
> which is the well known differential equation of a conic may be taken. It wants [means: It does not have] $x, \, y, \, y',$ and has multiple homogeneity, so may be depressed five orders in all, the result being an algebraic equation.
> 
> 
> 



> 
> **(end of the paper)** Hence the 5 depressions required in all may be effected in any of the following orders $[ \cdots]$ But this procedure is of course not so easy as that before shewn in abstract in Tables VIII. and IX., especially in the ascending steps, the reversal of Step II. not being so easy as in the former procedure. It is proposed to show the detail in a further paper.
> 
> 
> 


**[4]** Allan Joseph Champneys Cunningham (1842-1928)], [*Depression of differential equations*](https://books.google.com/books?id=bk6ZCcnSbrsC&pg=PA122), **Messenger of Mathematics** 18 (1889), 122-127. [JFM 20.0330.02](https://zbmath.org/20.0330.02)


[another link to the paper](https://gdz.sub.uni-goettingen.de/id/PPN599484047_0018)



> 
> **(first 2 sentences of the paper)** THIS paper is only a supplement to the paper on Depression of Differential Equations in pp. 118 to 145 of Vol. XVII. It contains the detail of three *additional* modes of solution (by depression) of the
> differential equation of a conic,
> $$9{y''}^{2}y^v \; - \; 45y''y'''y^{iv} \; + \; 40{y'''}^{3} \; = \; 0,$$
> not quite so simple as those given at pp. 140 to 144 of above.
> 
> 
> 


**[5]** [Asutosh Mukhopadhyay [Mookerjee] (1864-1924)](https://en.wikipedia.org/wiki/Ashutosh_Mukherjee), [*The geometric interpretation of Monge's differential equation to all conics*](https://books.google.com/books?id=YRfgAAAAMAAJ&pg=PA181), **Journal of the Asiatic Society of Bengal** 58 (Part 2) #2 (1889), 181-185.



> 
> **(first 1.5 pages of the paper)** Before proceeding to give the true geometric interpretation of Monge's differential equation to all conics, which I have recently discovered, and which it is the object of this paper to announce and establish, a brief survey of the past history and present position of the problem may not be wholly unprofitable. In the first place, then, we remark that the differential equation of all conics was, more than three-quarters of a century ago, first discovered by the illustrious French mathematician Gaspard Monge, and published by him in 1810.† {*footnote* † *states*: Sur les Equations différentielles des Courbes du Second Degré. (Bulletin de la Soc. Philom. Paris, 1810, pp. 87-88; Corresp. sur l'E'cole Polytech. (Hachette) Paris, 1809-13, t. ii, pp. 51-54).} It should be remembered that, in his paper, Monge does not furnish us with any clue to the method by which, from the integral equation of the conic, he derived the differential equation which now appropriately bears his name: neither is there any attempt at a geometric interpretation; it is simply stated that the differential equation to all conics of the second order as obtained by the elimination of the constants from the equation
> $$Ay^2 + 2Bxy + Cx^2 + 2Dy + 2Ex + 1 = 0$$
> is
> $$9q^{2}t \; – \; 45qrs \; + \; 40r^3 \; = \; 0,$$
> where, as usual,
> $$p = \frac{dy}{dx}, \;\;\; q = \frac{d^{2}y}{dx^2}, \;\;\; r = \frac{d^{3}y}{dx^3}, \;\;\; s = \frac{d^{4}y}{dx^4}, \;\;\; t = \frac{d^{5}y}{dx^5};$$
> and this statement is followed by a verification that the differential equation of all circles
> $$(1 + p^2)r \; = \; 3pq^2$$
> leads, on differentiation, to the differential equation of all conics.
> 
> 
> 



> 
> After Monge's paper, we come to the following statement made by the late Dr. Boole:‡ {*footnote* ‡ *is*: Differential Equations, Fourth Edition, pp. 19-20.} "Monge has deduced the general differential equation of lines [= curves] of the second order, expressed by the algebraic equation
> $$ax^2 + bxy + cy^2 + ex + fy = 1.$$
> It is
> $$9\left(\frac{d^{2}y}{dx^2}\right)^{2}\frac{d^{5}y}{dx^5} \;\; - \;\; 45\frac{d^{2}y}{dx^2}\frac{d^{3}y}{dx^3}\frac{d^{4}y}{dx^4} \;\; + \;\; 40\left(\frac{d^{3}y}{dx^3}\right)^3 \;\; = \;\; 0$$
> *But, here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms.*"
> 
> 
> 



> 
> The subject seems to have attracted the notice of English mathematicians from the above statement of Boole, and, during the thirty years which have elapsed since these remarks were first made, there appear to have been two attempts to interpret geometrically Monge's differential equation to all conics. The first of these propositions, by Lieut.-Col. Cunningham, is that the eccentricity of the osculating conic of a given conic is constant all round the latter.\* {*footnote* \* *is*: Quarterly Journal of Mathematics (1877), vol. xiv, pp. 226-229.} The second propositioin, by Prof. Sylvester, is that the differential equation of a conic is satisfied at the sextactic points of any given curve.† {*footnote* † *is*: American Journal of Mathematics (1886), vol. ix, pp. 18-19.} I have elsewhere considered in detail both these propositions, and I have fully set forth my reasons for holding that neither of them is the true geometric interpretation of Monge's defferential [sic] equation to all conics.‡ {*footnote* ‡ *is*: Journal, A. S. B. (1887), vol. lvi, part ii, pp. 134-145; P. A. S. B. (1887), pp. 185-186; P. A. S. B. (1888), pp. 74-86.}
> 
> 
> 


**[6]** [Edwin Bailey Elliott (1851-1937)](https://web.archive.org/web/20160409050248/http://www-groups.dcs.st-and.ac.uk:80/history/Biographies/Elliott.html), [*Note on the differential equation of a conic*](https://books.google.com/books?id=toZaAAAAYAAJ&pg=PA5), **Messenger of Mathematics** 19 (1890), 5-7. [JFM 21.0332.01](https://zbmath.org/21.0332.01)


[another link to the paper](https://gdz.sub.uni-goettingen.de/id/PPN599484047_0019)



> 
> **(first paragraph of the paper)** It is a surprisingly prevalent idea that the integration of
> $${y''}^{2}y^v \; - \; 5y''y'''y^{iv} \; + \; \frac{40}{9}{y'''}^{3} \; = \;0,$$
> the differential equation of a conic, is a matter of considerable difficulty. The recent papers† {*footnote* † *is*: Vol. XVII., pp. 118-145, Vol. XVIII., pp. 122-127.} of Colonel Cunningham on the depression of differential equations must have done much to dispel the notion; but his processes of depression and integration of the equation occupy, as he says, some considerable space when written at length. It may be worth while then to show how simple the matter really is by writing down two variations of a perfectly direct and elementary method of performing the integration.
> 
> 
> 


**[7]** [Francis Sowerby Macaulay (1862-1937)](https://web.archive.org/web/20190512182025/http://www-groups.dcs.st-and.ac.uk:80/%7Ehistory/Biographies/Macaulay.html) [journal editor], [[Solution to Problem 143](https://books.google.com/books?id=x5_xAAAAMAAJ&pg=PA94)], *Examination Questions and Problems* [column], **Mathematical Gazette** 1 #10 (February 1897), 94-95.


[JSTOR link to the item](https://www.jstor.org/stable/3604957)



> 
> Original statement of the problem in **Mathematical Gazette** 1 #9 (October 1896), p. 67. No attribution is given except an indication that the problem appeared on a recent Oxford University examination.
> 
> 
> 



> 
> **Statement of problem:** Prove that the result of eliminating the constants in the general equation of a conic by differentiation is
> $$\frac{d^3}{dx^3}\left\{\left(\frac{d^{2}y}{dx^2}\right)^{-\frac{2}{3}}\right\} = 0.$$
> 
> 
> 



> 
> **Published solution:** Let the equation of the conic be
> $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c \; = \; 0.$$
> Solving for $y,$ we have
> $$y \; = \; Dx + E \; \pm \; (Ax^2 + 2Bx + C)^{\frac{1}{2}},$$
> where $A,$ $B,$ $C,$ $D,$ $E$ are constants. Differentiating,
> $$\frac{dy}{dx} \; = \; D \; \pm \; \frac{Ax + B}{(Ax^2 + 2Bx + C)^{\frac{1}{2}}};$$
> and
> $$\frac{d^{2}y}{dx^2} \; = \; \pm \frac{A(Ax^2 + 2Bx + C) \; - \; (Ax + B)^2}{(Ax^2 + 2Bx + C)^{\frac{3}{2}}} \;\; = \;\; \pm \frac{AC \; - B^2}{(Ax^2 + 2Bx + C)^{\frac{3}{2}}};$$
> $$\therefore \;\; \left(\frac{d^{2}y}{dx^2}\right)^{-\frac{2}{3}} \;\; = \;\; (AC \; – \; B^2)^{-\frac{2}{3}}(Ax^2 + 2Bx + C);$$
> $$\therefore \;\;\frac{d^3}{dx^3}\left\{\left(\frac{d^{2}y}{dx^2}\right)^{-\frac{2}{3}}\right\} = 0.$$
> 
> 
> 


**[8]** [François Olivier Raoul Perrin (1841-1910)](https://www.annales.org/archives/x/raoulperrin.html), [*Sur quelques consequences géométriques de l'équation différentielle des coniques*](https://eudml.org/doc/86065) [On some geometrical consequences of the differential equation of the conics], **Bulletin de la Société Mathématique de France** 31 (1903), 54-64. [MR1504438](https://mathscinet.ams.org/mathscinet-getitem?mr=1504438); [JFM 34.0633.02](https://zbmath.org/34.0633.02)



> 
> This paper studies the differential equation $9(y'')^2y''''' – 45y''y'''y'''' + 40(y''')^3 = 0,$ which has all planar conic curves for its solutions.
> 
> 
> 


**[9]** François Olivier Raoul Perrin (1841-1910), [*Sur les intégrales de l'équation différentielle des coniques et leur interprétation géométrique*](https://eudml.org/doc/86059) [On the integrals of the differential equation of the conics and their geometric interpretation], **Bulletin de la Société Mathématique de France** 31 (1903), 275-285. [MR1504459](https://mathscinet.ams.org/mathscinet-getitem?mr=1504459); [JFM 34.0633.03](https://zbmath.org/34.0633.03)



> 
> This paper continues the investigation begun in Perrin's previous paper.
> 
> 
> 


