Existence of closed solutions for a polynomial first order differential equation
Consider the scalar differential equation \[ \frac{dx}{dt}= \sum^n_{k=0}a_k(t)x^k, \tag{*} \] where \(a_k\in C([0,2\pi])\), \(k=0, \dots,n\), \(n\geq 2\). Assuming \(\int^{2 \pi}_0a_1 (t)dt\neq 0,\) the author derives conditions for (*) to have at least one solution satisfying \(x(0)= x(2\pi)\). He applies his result to planar polynomial vector fields to get criteria for the existence of at least one limit cycle.
scientific article; zbMATH DE number 5124017