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 $\mathfrak{z}=0$. Periodic solutions with polynomial coefficients are executed for classes $C_{10,3},C_{10,4},$ and $C_{10,5}.$ Limit cycles are found for both non-homogeneous and homogeneous polynomials with trigonometric coefficients for classes $C_{22,11}$,$C_{24,12}$ and $C_{10,10}$, respectively. We developed a formula ${\varkappa }_{10}$, which is not available in literature. By using our newly developed formula, we succeeded to find highest known multiplicity 10 for the classes $C_{10,3},C_{10,4}$ with algebraic and $C_{10,10},C_{24,12}$ 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.

本文讨论了常微分方程周期解数的发展。我们从精细焦点出发研究了周期解的一阶非自治微分方程的焦点值$\mathfrak{ž}=0$。对类执行具有多项式系数的周期解$C_{10，3}，C_{10，4}，$ 和 $C_{10，5}。$ 发现非三角多项式和齐次多项式的极限环 $C_{22，11}$，$C_{24，12}$ 和 $C_{10，10}$， 分别。我们制定了一个公式${\varkappa }_{10}$，这在文献中没有。通过使用我们新开发的公式，我们成功地找到了类别的已知最高多重性10$C_{10，3}，C_{10，4}$ 与代数和 $C_{10，10}，C_{24，12}$ 与三角系数。我们提出了各种多项式类别及其分叉分析，这证实了所提出方法的一般性和真实性。