First Integrals of One Fourth Order Ordinary Differential Equation

Abstract

   First integrals for one fourth-order ordinary differential equation that is a special case of one of the higher analogues of the Painlevé equation have been found. The equation has been studied for the presence of the Painlevé property using the Kovalevskaya algorithm. It has been shown that the method used does not allow to accurately determining whether the general solution of the equation has critical moving singular points or not. Two particular cases of the equation under study with certain values of its parameters contained have been considered in detail. To find the first integrals of the resulting equations, their linear dependence on the highest derivative is assumed. The found first integrals are used to reduce the order of the equations under study. This equation does not have any first integral of a similar form. The found first integral of this equation is used to reduce the equation to a second order equation. It has been shown that the resulting equation does not have a first integral of the form similar to the cases of the third and fourth order equations.