The quasilinear second order differential equation .. x + (1 + p (x, t)) . x + p (x, t) x = f (x, t) where p, f ∈ C (D,R) , D = Ix×I, Ix ⊆ R open set, I = 〈 t,∞ 〉 is under consideration. The paper presents some results on the existence and behavior of parameter classes of solutions of this equation. The qualitative analysis theory and topological retraction method are used. The general results are presented and subsequently certain examples are considered.

This paper deals with certain classes of Cauchy’s solutions of quasilinear second order differential equations in general form, Van der Pol's differential equation, which is used in the theory of electric circuits, and Lagerstorm's differential equations, which is used in asymptotic treatment of viscous flow past a solid at low Reynolds number. Behaviour of integral curves in the neighbourhoods of an arbitrary or integral curve is considered. Obtained results establish sufficient conditions for the existence and asymptotic behaviour of the observed equations. The obtained results contain the answer to the question on approximation of solutions whose existence is established. The errors of the approximation are defined by functions that can be sufficiently small. The qualitative analysis theory and topological retraction methods were used. © 2014 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of DAAAM International Vienna.

The aim of this paper is to establish the existence, behavior and approximation of certain classes of solutions of the quasilinear systems of differential equations. Behavior of integral curves in neighborhoods of an arbitrary or certain curve is considered. The approximate solutions with precise error estimates are determined. The theory of qualitative analysis of differential equations and topological retraction method are used.

The present paper deals with the nonlinear systems of differential equations of Volterra type regarding to the existence, behaviour, approximation and stability of their definite solutions, all solutions in a corresponding region or parametric classes of solutions on unbounded interval. The approximate solutions with precise error estimates are determined. The theory of qualitative analysis of differential equations and topological retraction method are used.

This paper deals with the behavior, approximation and stability of some solutions of the system of quasilinear differential equations. The behavior of solutions in the neighbourhood of an arbitrary curve is considered, with extraordinary attention on some special cases. The obtained results contain an answer to the question on approximation as well as stability of solutions whose existence is established. The errors of the approximation are defined by the function that can be sufficiently small. The theory of qualitative analysis of differential equations and topological retraction method are used.