TY  - JOUR
ID  - Tsyfra2021
LB  - Tsyfra2021
AU  - Tsyfra, Ivan
TI  - On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations
ST  - On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 5
SP  - 685
EP  - 699
DO  - https://doi.org/10.7494/OpMath.2021.41.5.685
UR  - https://doi.org/10.7494/OpMath.2021.41.5.685
AB  - We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.
KW  - ordinary differential equation
KW  - partial differential equation
KW  - integrability
KW  - symmetry
KW  - quadrature
KW  - Lie transformation group
SN  - 1232-9274
ER  -