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 [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

[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)

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)