On Liouvillian integrability of the first–order polynomial ordinary differential equations
Copyright policy )On Liouvillian integrability of the first–order polynomial ordinary differential equations
The authors prove the following result: Theorem. If a complex differential equation of the form \[ {dy\over dx}= a_0(x)+ a_1(x) y+\cdots+ a_n(x) y^n, \] where \(a_i(x)\), \(i= 0,\dots, n\), are polynomials in \(x\), \(a_n(x)\neq 0\), \(n\geq 2\), has a Liouvillian first integral, then it has a finite invariant algebraic curve.
Abel differential equation, Applied Mathematics, Explicit solutions, first integrals of ordinary differential equations, Riccati differential equation, invariant algebraic curve, Liouvillian integrability, Invariant manifolds for ordinary differential equations, Analysis, Invariant algebraic curve, Ordinary differential equations in the complex domain
Abel differential equation, Applied Mathematics, Explicit solutions, first integrals of ordinary differential equations, Riccati differential equation, invariant algebraic curve, Liouvillian integrability, Invariant manifolds for ordinary differential equations, Analysis, Invariant algebraic curve, Ordinary differential equations in the complex domain
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