Jacobi equation

From Encyclopedia of Mathematics
Revision as of 19:27, 31 March 2017 by Ivan (talk | contribs) (TeX)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A first-order ordinary differential equation

$$\frac{dy}{dx}=\frac{Axy+By^2+ax+by+c}{Ax^2+Bxy+\alpha x+\beta y+\gamma}$$

or, in a more symmetric form,



where all the coefficients are constant numbers. This equation, which is a special case of the Darboux equation, was first studied by C.G.J. Jacobi [1]. The Jacobi equation is always integrable in closed form by using the following algorithm. First one finds by direct substitution at least one particular linear solution


Then one makes the changes of variables

$$\xi=\frac x{px-y+q},\quad\eta=\frac y{px-y+q},$$

to obtain an equation that is reducible to a homogeneous equation.


[1] C.G.J. Jacobi, "De integratione aequationis differentialis $(A+A'x+A''y)(x\partial y-y\partial x)-(B+B'x+B''y)\partial y+(C+C'x+C''y)\partial x=0$" J. Reine Angew. Math. , 24 (1842) pp. 1–4
[2] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)



[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Jacobi equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article