The solvability conditions for the second order nonlinear differential equation with unbounded coefficients in L2(R)

The article deals with the existence of a generalized solution for the second order nonlinear differential equation in an unbounded domain. Intermediate and lower coefficients of the equation depends on the required function and considered smooth. The novelty of the work is that we prove the solvability of a nonlinear singular equation with the leading coefficient not separated from zero. In contrast to the works considered earlier, the leading coefficient of the equation can tend to zero, while the intermediate coefficient tends to infinity and does not depend on the growth of the lower coefficient. The result obtained formulated in terms of the coefficients of the equation themselves; there are no conditions on any derivatives of these coefficients.

Research of a third-order nonlinear differential equation in the vicinity of a moving singular point for a complex plane

This article generalizes the previously obtained results of existence and uniqueness theorems for the solution of a third-order nonlinear differential equation in the vicinity of moving singular points in the complex domain, as well as constructs an analytical approximate solution, and obtains a priori estimates of the error of this approximate solution. The study was carried out using the modified method of majorants to solve this equation, which differs from the classical theory, in which this method is applied to the right-hand side of the equation The final point of the article is to conduct a numerical experiment to test the theoretical positions obtained.

Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

In Lorentz-Minkowski 3-space, null scrolls are ruled surfaces with a null base curve and null rulings. Their mean, as well as their Gaussian curvature, depends only on a parameter of a base curve. In the present paper, we obtain the first-order nonlinear differential equation (Riccati equation) which relates curvatures of a base curve to curvatures of a null scroll. Conditioned by this equation, we can determine a family of null scrolls with a given null base curve and prescribed curvatures, in particular, a family of minimal and constant mean curvature null scrolls.

On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation

In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An example is provided to illustrate the main results.

Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

In this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.