"分离变量法解析不同坐标系下的齐次和非齐次偏微分方程"

Partial differential equations (PDEs) play a crucial role in mathematics and serve as an essential foundation for many other disciplines. As a result, the study and exploration of methods to solve PDEs have always been significant areas of interest and research for scientists. Among the various methods used to solve PDEs, the most straightforward and widely used approach is the method of separation of variables. This article combines the conditions and requirements for applying the method of separation of variables and provides a comprehensive analysis of various aspects such as whether the PDE is homogeneous, whether the boundary conditions are homogeneous, and whether the boundary shape is regular. The method of separation of variables is then applied to solve the PDE, and the process of solving PDEs using this method is thoroughly explained. In addition, the article also explores the solution of PDEs in different coordinate systems. Due to the wide-ranging applications of PDEs in almost every field and aspect, and with the continuous expansion of various branches in different disciplines, the scope of applications for PDEs will continue to grow. Therefore, there is still a lot of ongoing research required to solve PDEs. In summary, this article highlights the importance of solving PDEs and the significant efforts made by scientists in exploring and studying solution methods. The method of separation of variables is presented as a simple and effective approach for solving PDEs. The article provides a comprehensive analysis of the conditions and requirements for applying this method and demonstrates its application in solving PDEs in various coordinate systems. However, given the vast and expanding applications of PDEs, further research is needed to tackle the challenges and complexities that arise in solving them. Keywords: separation of variables, partial differential equations, homogeneous, non-homogeneous.