Abstract
This article deals with the development of the number of periodic solutions for ordinary differential equations. We investigated focal values for first-order non-autonomous differential equation for periodic solutions from a fine focus . Periodic solutions with polynomial coefficients are executed for classes and Limit cycles are found for both non-homogeneous and homogeneous polynomials with trigonometric coefficients for classes , and , respectively. We developed a formula , which is not available in literature. By using our newly developed formula, we succeeded to find highest known multiplicity 10 for the classes with algebraic and with trigonometric coefficients. We present a variety of polynomial classes along with their bifurcation analysis which confirms the generality and authenticity of the method presented.
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