Solutions of ordinary differential equations with Lie and Noether symmetries

In this thesis, Lie and Noether symmetries and solutions of ordinary differential equations are focused on. Differential equations are widely used in formulating many technological problems as well as many of the fundamental laws of nature. Sometimes it is possible to integrate equations when traditional and problem-specific methods are used to obtain the analytical solution of the equations. Most of the time it is not possible to integrate these equations. There are more than four hundred integrable ordinary differential equations in the literature, and each of them has its own unique solution methods. Lie group analysis provides a much more general solution method by reducing all these ordinary differential equations to four different types, and therefore Lie theory appears as a very important tool in obtaining the solution of differential equations. Another tool that plays an important role in the analysis of differential equations is the Noether symmetry method. For this method, Noether's theorem and first integrals corresponding to first and second order Lagrangians are given. The symmetries of the differential equations are used to obtain the first integrals. In this thesis, Noether's theorem is used to obtain the first integrals corresponding to first and second order Lagrangians. Noether proved that for differential equations there is a first integral corresponding to every symmetry generator, obtained from the variation principle. These symmetries are called Noether symmetry generators, and if a Noether symmetry generator exists, Noether's theorem easily provides the first integrals corresponding to each symmetry generator. Lie theory and Noether's theorem involve determining the symmetries of a system and then finding the corresponding invariants of its constants. While the differential equation is left invariant in the case of Lie theory, Noether's theorem leaves the Action Integral invariant. Obtaining these makes it easier to solve the equations because it reduces the order of differential equations. Based on this, within the scope of the thesis, Lie and Noether symmetry methods were analyzed with the Emden-Fowler equation and ordinary differential equations, and how the obtained data can be used to obtain the solution of the given equation is shown in an explanatory manner on various examples. Thus, it is discussed how to find solutions using Lie and Noether symmetry methods for some nonlinear equations that cannot be solved by known methods due to their unique structure.