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Nonlinear differential equations are considered to be an important tool for describing a number of phenomena in engineering and the natural sciences, and their study is thus subject to contemporary research. The purpose of the paper is to show applications of the differential transform to second-order half-linear Euler equations with and without delay. The case of proportional delay is considered. Finding a numerical solution to an initial value problem is reduced to solving recurrence relations. The outputs of the recurrence relations are coefficients of the Taylor series of the solution. Validity of the presented algorithm is demonstrated on concrete examples of initial value problems. Numerical results are compared with solutions produced by Matlab function "ddesd". • Introduction to second-order half-linear Euler equations. • Description of the differential transform in combination with method of steps. • Application of the differential transform algorithm to half-linear Euler equation. • Comparison of the numerical results to those produced by Matlab function "ddesd". [ABSTRACT FROM AUTHOR]

Abstract: We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation

We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = |x|^{p - 1} \operatorname{sgn} x,p > 1, $$ where r, c are continuous functions and r(t) > 0.