# Equivalence and symmetries of first order differential equations

• Volume: 58, Issue: 3, page 605-635
• ISSN: 0011-4642

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## Abstract

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In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations $\overline{x}=\varphi \left(x\right),$$\overline{y}=\overline{y}\left(\overline{x}\right)=L\left(x\right)y\left(x\right).$ That means, the transformed unknown function $\overline{y}$ is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind ${F}^{j}={a}_{j}y\Pi |{z}_{i}{|}^{{k}_{i}^{j}}={a}_{j}y|{z}_{1}{|}^{{k}_{1}^{j}}...|{z}_{m}{|}^{{k}_{m}^{j}}={a}_{j}\left(x\right)y|y\left({\xi }_{1}\right){|}^{{k}_{1}^{j}}...{|y\left({\xi }_{m}\right)|}^{{k}_{m}^{j}}$ is compared to similar results obtained by means of auxiliary functional equations.

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