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 presents sufficient conditions for the existence of solutions for certain classes of Cauchy’s solutions of the Lagerstrom equation as well as their behavior. Behavior of integral curves in the neighborhoods of an arbitrary or integral curve are considered. 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 theory of qualitative analysis of differential equations and topological retraction method are used.

Abstract Laser beam machining (LBM) is one of the most widely used thermal energy based non-contact type advance machining process which can be applied for almost whole range of materials. This paper defines mathematical models for surface roughness prediction (Ra, μm) and width of heat affected zone (HAZ, mm) during laser cutting of alloy steels 1.4571 and 1.4828 with nitrogen as assist gas. For defining appropriate mathematical models multiple regression analysis is used with four independent variables. Following parameters are varied: cutting speed, focus position, nitrogen assist gas pressure and stand-off. Obtained mathematical models describe dependence of Ra and HAZ from varied process parameters.

Abstract 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.