Solving Ordinary Differential Equations with Variable Coefficients by Complex Al-Zughair Transform

In this research, a special type of ordinary differential equations called Ali's equation is solved, which is in the following form \begin{equation*} a_0(\ln ({\unicode{x1D75}}))^n \frac{d^n {\unicode{x024F}}}{d x^n}+a_1(\ln ({\unicode{x1D75}}))^{n-1} \frac{d^{n-1} {\unicode{x024F}}}{d x^{n-1}}+\cdots+a_{n-1}(\ln ({\unicode{x1D75}})) \frac{d {\unicode{x024F}}}{d x}+a_n {\unicode{x024F}}=f({\unicode{x1D75}})\end{equation*} using the complex AI-Zughair transform. Which leads to opening new horizons in this field, especially in physical, mechanical and other applications. We find the convolution of the transform, afterwards we solve new types of ordinary and partial differential equations.