# Jacobi equation

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,

$$(a_1x+b_1y+c_1)(xdy-ydx)+{}$$

$${}+(a_2x+b_2y+c_2)dx-(a_3x+b_3y+c_3)dy=0,$$

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 . 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

$$y=px+q.$$

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.

How to Cite This Entry:
Jacobi equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_equation&oldid=40763
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article