Recently, the variable coefficient homogeneous differential equations (VCHDE) have been widely applied to real-world problems, such as wave propagation and material science. However the exploration and research on higher-order VCHDE is relatively lagging. Given this, this work focuses on the solutions of fourth-order and nth-order VCHDE with polynomial coefficients. By means of the sufficient conditions for the existence of solutions to differential equations, a connection is established between the rank of the variable coefficient matrix and the existence of polynomial particular solutions. The main results show that: (1) the necessary and sufficient conditions for the existence of polynomial particular solutions of fourth-order VCHDE are derived; (2) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of fourth-order VCHDE are proved; (3) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of nth-order VCHDE are proved. These results not only extend the class of solvable differential equations, but also provide a new way of thinking about the existence of solutions to VCHDE.

In this paper, we investigate the existence and structure of power-type solutions for Caputo fractional differential equation systems (CFDESs) and Grünwald–Letnikov fractional differential equation systems (GLFDESs). Building on the definitions of the Caputo fractional derivative (CFD) and the Grünwald–Letnikov fractional derivative (GLFD), we derive explicit expansion formulas for the fractional differential operators, construct joint coefficient-solution matrices for the considered systems, and, from these, obtain necessary and sufficient rank conditions for the existence of $m$-th order power solutions. On this basis, we further (1) establish equivalent criteria that guarantee the uniqueness of the degree of power solutions and (2) derive rank-based conditions for the existence of two, and more generally $p$, linearly independent power particular solutions with distinct degrees. Taken together, these results provide a unified matrix-based theoretical framework for analyzing the existence, uniqueness, and multiplicity of power-type solutions and the associated system structure of the two types of fractional differential systems. Two numerical examples are also provided to demonstrate the validity of the proposed results.