# 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.

## How to cite

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Tryhuk, V.. "Equivalence and symmetries of first order differential equations." Czechoslovak Mathematical Journal 58.3 (2008): 605-635. <http://eudml.org/doc/37856>.

@article{Tryhuk2008,
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\}=\varphi (x),$$\bar\{y\}=\bar\{y\}(\bar\{x\})=L(x)y(x). That means, the transformed unknown function \bar\{y\} is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind F^j=a_j y \Pi |z_i|^\{k^j_i\}=a_j y |z_1|^\{k^j_1\} \ldots |z_m|^\{k^j_m\}=a_j(x)y|y(\xi _1)|^\{k^j_1\}\ldots |y(\xi _m)|^\{k^j_m\} is compared to similar results obtained by means of auxiliary functional equations.}, author = {Tryhuk, V.}, journal = {Czechoslovak Mathematical Journal}, keywords = {differential equations with deviations; equivalence of differential equations; symmetry of differential equation; differential invariants; moving frames; differential equations with deviations; equivalence of differential equations; symmetry of differential equation; differential invariants; moving frames}, language = {eng}, number = {3}, pages = {605-635}, publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic}, title = {Equivalence and symmetries of first order differential equations}, url = {http://eudml.org/doc/37856}, volume = {58}, year = {2008}, } TY - JOUR AU - Tryhuk, V. TI - Equivalence and symmetries of first order differential equations JO - Czechoslovak Mathematical Journal PY - 2008 PB - Institute of Mathematics, Academy of Sciences of the Czech Republic VL - 58 IS - 3 SP - 605 EP - 635 AB - In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations \bar{x}=\varphi (x),$$\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.
LA - eng
KW - differential equations with deviations; equivalence of differential equations; symmetry of differential equation; differential invariants; moving frames; differential equations with deviations; equivalence of differential equations; symmetry of differential equation; differential invariants; moving frames
UR - http://eudml.org/doc/37856
ER -

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