Equivalence and symmetries of first order differential equations

Tryhuk, V.

Czechoslovak Mathematical Journal, Tome 58 (2008) no. 3, pp. 605-635.

Résumé

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 $\bar x=\varphi (x),$ $\bar y=\bar y(\bar x)=L(x)y(x).$ 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 $F^j=a_j y \Pi |z_i|^{k^j_i}=a_j y |z_1|^{k^j_1} \ldots |z_m|^{k^j_m}=a_j(x)y|y(\xi _1)|^{k^j_1}\ldots |y(\xi _m)|^{k^j_m}$ is compared to similar results obtained by means of auxiliary functional equations.

Détail

Classification : 34A30, 34A34, 34K05, 34K17
Keywords: differential equations with deviations; equivalence of differential equations; symmetry of differential equation; differential invariants; moving frames