The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent...

Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\overline{x}=\varphi \left(x\right)$, $\overline{y}=L\left(x\right)y$, $\overline{z}=M\left(x\right)z,...$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\text{'}}=f(x,y,z,z^{\text{'}})$) and certain differential equations with deviation (if $z=y\left(\xi \right(x\left)\right)$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the...