On asymptotic properties of solutions, defined on the half of axis of one semilinear ODE

The paper deals with the solutions of ordinary differential semi-linear equation, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden–Fowler equation. Asymptotic behavior of Emden–Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order derivative term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I. T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.