Connections between symmetries and integrating factors of ODEs

In this thesis we examine the connections between conservation laws and symmetries, both for self-adjoint and non self-adjoint ODEs. The goal is to gain a better understanding of how to combine symmetry methods with the method of conservation laws to obtain results not obtainable by either method separately. We review the concepts of symmetries and integrating factors. We present two known methods of obtaining conservation laws without quadrature, using known conservation laws and symmetries. We show that the two methods yield the same result. For self-adjoint systems, we examine Noether's theorem in detail and discuss its generalisation for ODEs admitting more than one variational symmetry. We generalise an example from Sheftel [20] and show how to use r-dimensional Lie Algebra of variational symmetries to obtain more then r reductions of order. We develop an ansatz for finding point variational symmetries. We also develop ansatzes that use a known symmetry to find an integrating factor or another symmetry. These ansatzes are then used to classify ODEs. New solvable cases of Emden-Fowler and Abel ODEs result.Science, Faculty ofMathematics, Department ofGraduat