الفهرس 
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Abstract
First integral of ordinary differential equations (ODEs) is one of the important methods for reducing the order of ODEs and also study the behavior of the solutions of the ODEs. Each first integral leads to a reduction of the order of the ODE without change of variables and this is different from symmetry reduction which relies on a change of variables.
The aim of these the thesis is to investigate some methods used for obtaining the first integral of the differential equations. In addition, we use these methods to obtain some exact solutions for some engineering mathematical models such as fin equations and some nonlinear evolution equations. The thesis is organized in six chapters : In Chapter 1, we present a brief introduction to the basic concepts of Lie group analysis and how to find Lie point symmetries of differential equations. Also, we show how to reduce the order of the ordinary differential equations or reduce the number of the independent variables in case of the partial differential equations using Lie point symmetry method. Finally, we introduce some engineering applications which are solved using Lie point symmetry method such as fin equation and modified KdV Equation. In chapter 2, we introduce some concepts such as adjoint equations, self adjoint equation and formal Lagrangian. Also, the first integral (conservation law) of ODE is introduced. Then, we introduce partial Lagrangian of ODE and how use it to obtain the first integral. Finally, we introduce fin problem which is solved using partial Noether method. In chapter 3, we introduce integrating factor method and how to use it to reduce the order of the ODE. Then, we investigate the relation between the first integral and integrating factor method. Finally, we introduce optical metamaterials problem which is solved using integrating factor method. In chapter 4, we introduce the linearization method of the ODEs and how to use it to linearization the ODEs. Then, we show the relation between the first integral and linearization method of the ODE. Finally, we introduce the Kudryashov and Sinelshchikov equation which is solved using the linearization method of the ODEs. In chapter 5, we introduce some methods for determining the Lagrangian of the differential equation and the relation between the first integral and Lagrangian of the differential equation. In chapter 6, we give the conclusion and suggested future work.