**The moving frames for differential equations**

**II. Underdetermined and functional equations**
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*Vaclav Tryhuk, Oldrich Dlouhy*

**Address.**

Department of Mathematics, Faculty of Civil Engineering

Brno University of Technology

Veveri 331/95, 662 37 Brno, Czech Republic

**E-mail. **tryhuk.v@fce.vutbr.cz**
**dlouhy.o@fce.vutbr.cz

**Abstract.**

Continuing the idea of Part I, we deal with more involved

pseudogroup of transformations $\bar x=\varphi (x)$, $\bar

y=L(x)y$, $\bar z=M(x)z,\, \ldots$ applied to the first order

differential equations including the underdetermined case

(i.e. the Monge equation $y'=f(x,y,z,z')$) and certain differential

equations with deviation (if $z=y(\xi (x))$ 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 method of moving frames adapted to the theory of differential

and functional-differential equations.

**AMSclassification. **34-02, 34K05, 34A30, 34A34, 34K15.

**Keywords.** Pseudogroup, moving frame, equivalence of differential

equations, differential equations with delay.