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Liouvillian first integrals of second order polynomial differential equations. (English) Zbl 0939.34002

The author proves the following theorem: If the system \[ \dot x= P(x,y),\quad \dot y= Q(x,y)\quad (P,\;Q\text{ polynomials})\tag{1} \] has a Liouvillian integrating factor of the form: \[ \exp\{\int Udx+ Vdy\},\quad U_y= V_x,\tag{2} \] where \(U\), \(V\) are rational functions of \(x\) and \(y\), then there exists a Darbouxian integrating factor of the form: \[ \exp(D/E)\Pi C^{\ell_i}_i,\tag{3} \] where \(D\), \(E\), \(C_i\) are polynomials in \(x\) and \(y\).
(3) defines a collection of invariant algebraic curves of (1), \(C_i(x,y)= 0\), satisfying the equation: \[ {d\over dt} C_i(x, y)= C_i(x, y)L_i(x,y)\tag{4} \] for some polynomial \(L_i(x,y)\) of smaller degree than that of \(P\), \(Q\), and \(\exp(D/E)\) satisfies a similar equation. Thus, the search for (2) can be reduced to the search for invariant algebraic curves and exponential factors of (1) satisfying (4).

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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