On the lie symmetries of first order ordinary differential equations

This thesis consists of three chapters. In the first chapter, we discuss how to find Lie symmetry transformations of the first-order ordinary differential equations and how to obtain the solutions of the ordinary differential equations by using these Lie symmetry transformations.In the second chapter, we mention about finding the Lie symmetries of the second-order ordinary differential equations systematically unlike the first-order. And we give a brief summary about Lie algebra that is used to obtain solutions of the ordinary differential equations with their Lie symmetries.In the third chapter, we consider second-order ordinary differential equations derived from the original first-order ordinary differential equations and also first-order linear partial differential equations obtained by writing characteristic into the linearized symmetry condition in order to resolve an important deficiency for the first-order ordinary differential equations which have no systematic way to find their Lie symmetries.