Generalized cofactors and nonlinear superposition principles
In this note, the authors show that the Darboux theory of integrability can be applied to some planar polynomial vector fields \(\dot x=X(x)\) having enough invariant curves, which are not necessarily polynomial. Recall that if \(Xf=kf\) for some smooth function \(k,\) then \(f=0\) is an invariant solution of \(X\) and \(k\) is called the cofactor of \(f.\) They prove that if \(X(x)\) has degree \(m\) and if the system \(\dot x= X(x)\) possesses \(m(m+1)/2\) invariant curves \(f_i(x,y)=0,\) all of them with associated polynomial cofactors of degree \(m-1\), then \(X\) has either a first integral or an integrating factor, which can be constructed from these curves. The following illustrative example is given. The planar differential equation \(\dot x=-y+x^4,\) \(\dot y=x,\) has the invariant curves \(f_1(x,y)=2x^2{Ai}(4^{1/3}y)+{Ai}'(4^{1/3}y)=0 \) and \(f_2(x,y)=2x^2{Bi}(4^{1/3}y)+{Bi}'(4^{1/3}y)=0, \) with cofactors \(k_1(x,y)=k_2(x,y)=2x^3,\) where \({Ai}\) and \({Bi}\) are a pair of linearly independent solutions of \(w''(z)=zw(z).\) The system possesses the non-Liouvillian first integral \(H(x,y)=f_1(x,y)/f_2(x,y).\)
planar polynomial differential equations
scientific article; zbMATH DE number 2073826