Conservation laws, first integrals, integration factors, symmetry group classifications and similarity solutions of differential equations

In this study, first of all, some methods that allow to find the first integrals and integration factors of the differential equations are explained in detail and these methods are applied to Li\'{e}nard I-type and Li\'{e}nard II-type equations. The relationship between the methods that are studied here are analyzed then their indirect and direct connections between each other are evaluated. The important relations that are $\lambda$-symmetries with Lie symmetries, Prelle-Singer method with $\lambda$-symmetries and Lie symmetries with Jacobi multipliers are claimed. Noether symmetries, $\lambda$-symmetries via Lie symmetries, integration factors, first integrals and Jacobi multipliers of Li\'{e}nard I-type and Li\'{e}nard II-type equations are derived. The method of equivariant moving frame has been introduced, evaluation of this method as a new point of view has been discussed and then explained how it is applied. With the goal of creating better understandig, for the some basic group actions equivariant moving frames has been constructed. Differential and joint differential invariants of binary and ternary forms are performed with respect to the moving frame method. Furthermore, higher order differential invariant of ternary forms are constructed and we show that generating invariant of the differential invariant algebra of ternary forms.