Abstract
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 3 = 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, C(10,5 )and C-12,C-6 with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C-8,C-3 and C-8,C-4 with algebraic coefficients have at most eight limit cycles. The new formula x io is developed by which we succeeded to find highest known multiplicity ten for class C-9,C-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.