Electronic Journal of Differential Equations (Oct 2017)
Trigonometric polynomial solutions of equivariant trigonometric polynomial Abel differential equations
Abstract
Let $A(\theta)$ non-constant and $B_j(\theta)$ for $j=0,1,2,3$ be real trigonometric polynomials of degree at most $\eta \ge 1$ in the variable x. Then the real equivariant trigonometric polynomial Abel differential equations $A(\theta) y' =B_1(\theta) y +B_3 (\theta) y^3$ with $B_3 (\theta)\ne 0$, and the real polynomial equivariant trigonometric polynomial Abel differential equations of second kind $A(\theta) y y' = B_0(\theta)+ B_2(\theta) y^2$ with $B_2 (\theta)\ne 0$ have at most 7 real trigonometric polynomial solutions. Moreover there are real trigonometric polynomial equations of these type having these maximum number of trigonometric polynomial solutions.
