Reduction of differential equations with symmetries and obtaining their exact solutions

In this master's thesis, the reduction of differential equations with the help of symmetries and obtaining their exact solutions are examined in detail. Differential equations, which are used in mathematical physics, engineering, biology and many other branches of science, are powerful tools for modeling complex processes and systems. However, solving these equations usually involves analytical difficulties, and this thesis deals with the use of symmetry and exact solution techniques to overcome these difficulties. The first part of the study will emphasize a general definition of differential equations and their importance. Understanding how differential equations appear in nature and engineering applications forms the basis of the study. In the second part, the role of the concept of symmetry in mathematics will be discussed, and how differential equations can be reduced using symmetries will be explained. The third part will focus on obtaining exact solutions to differential equations. The fourth chapter discusses the reduction of orders of differential equations through symmetry analysis, explaining methods for reducing higher-order equations to lower orders using Lie symmetries. The fifth chapter evaluates the application of theoretical methods, presenting exemplary models in a wide range of fields, from mechanical systems and electrical circuits to quantum mechanics, epidemiology, and economics. The study concludes with a summary of the findings, a discussion of the benefits of reductions using symmetries, and the use of exact solutions of differential equations.