Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation
Second-order ordinary differential equations invariant under translation \(\partial_x\) in the independent variable and rescaling \(-qx \partial_x +y \partial_y\) are studied from the Lagrangian point of view. The general form of such equation is \[ \frac{y''}{y^{2q+1}} +f(\frac{y'}{y^{q+1}})=0. \tag{1} \] The authors present a method to determine the Lagrangian of equation (1) and to analyze the equations in terms of the Painlevé analysis and their Lagrangians from the Noetherian approach. The establishment of a relationship between the Euler-Lagrange equations and their Lagrangians provides a general method for dealing with equations of the class under consideration. The Lagrangian of equation (1) is represented in the form \(L=y^n F(y'/y^2)\) since the variable \(y'/y^2\) appears as an arbitrary argument in the second-order differential equation. The multiplier of \(F\) in the expression of the Lagrangian can be chosen to be any function of \(y\) or \(y'\). However, the differential equation for \(F\) imposes an effective restriction on the form of the multiplier of \(F\) due to the problems of integrating the equation. The associated Euler-Lagrange equation is \((2u^2+f(u))F'' -nuF'+nF=0\) with \(u=y'/y^2\). The coefficient of \(F''\) in this equation is taken to be \(2u^2+f(u)=m(u+\alpha)^2+\beta\). A feature associated with the class of differential equations for \(F\) resulting from the choice of a quadratic \(f=f(u)\) is that in general they are hypergeometric functions. The Painlevé analysis of the equations arising in the different cases is performed.
second-order ordinary differential equations
scientific article; zbMATH DE number 2173084
scientific article