Equivalence and symmetries of first order differential equations

Abstract

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 $\bar x = \phi (x), \bar y = \bar y(\bar x) = L(x)y(x)$. That means, the transformed unknown function $\bar y$ 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 $F^j = a_j y \prod \left| {z_i } \right|^{k_i^j } = a_j y\left| {z_1 } \right|^{k_1^j } \ldots \left| {z_m } \right|^{k_m^j } = a_j (x)y\left| {y(\xi _1 )} \right|^{k_1^j } \ldots \left| {y(\xi _m )} \right|^{k_m^j }$ is compared to similar results obtained by means of auxiliary functional equations.