Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus z = 0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, C10, 5 and C12, 6 with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C8, 3 and C8, 4 with algebraic coefficients have at most eight limit cycles. The new formula k10 is developed by which we succeeded to find highest known multiplicity ten for class C9,3 with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.