Published electronically January 10, 2023

DOI: 10.1137/21S1454110

**Authors:** Feng Jiang (University of Nottingham Ningbo China), Zhengyang Guo (University of Nottingham Ningbo China), Dongâ€™ang Liu (University of Nottingham Ningbo China), and Yanghao Wang (University of Nottingham Ningbo China) **Project Advisors:** Behrouz Emamizadeh (University of Nottingham Ningbo China) and Amin Farjudian (University of Nottingham Ningbo China)

**Abstract:** This note is concerned with the qualitative properties of the solutions of a class of linear ordinary differential equations. The existence and uniqueness of solutions are addressed, and properties of the graph of the solution when imposing some restrictions are derived. A new notion of derivative, called the force derivative, is introduced and an orthogonality result, between the force derivative of the solution and the force function, is obtained. All the important results are verified by numerical examples using MATLAB. Finally, an inequality result reminiscent of the famous G. Talenti's inequality is proved.