View source for Jacobi equation

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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|Darboux equation]], was first studied by C.G.J. Jacobi [[#References|[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

$$y=px+q.$$

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,   "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>