Subfunctions for second order ordinary differential equations

Explicit upper and lower bounds are constructed for a functional defined on the difference of solutions to a class of nonlinear hyperbolic problems modelling string vibrations. Such bounds indicate how solutions are affected by the input data over a finite time interval. The logarithmic convexity method is used to obtain two second order ordinary differential inequalities for the appropriate functional. These inequalities are then related to two systems of first order equations whose solutions are the desired bounds.     

On the numerical solution of second order ordinary differential equations in the high-frequency regime

We describe an algorithm for the numerical solution of second order linear ordinary differential equations in the high-frequency regime. It is based on the recent observation that solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate this and other properties of the method with several numerical experiments.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

On periodic solutions of certain second order ordinary differential equations with periodic coefficients

Acta Mathematica Hungarica -     

Linear ordinary differential equations of the second order with unbounded solutions

We consider three families of equations of the form y″ + (1 + φ(x))y = 0, where the coefficient φ(x) satisfies the condition lim _x→+∞ φ(x) = 0. We obtain solutions of these equations in closed form. We show that the maximum absolute values of solutions grow at the rate of a logarithmic function, a power-law function, and even an exponential function as x → ∞.     

Neumann problems for second order ordinary differential equations across resonance

This paper deals with the existence-uniqueness problem to Neumann problems for second order ordinary differential equations probably across resonance. By the optimal control theory method, some global optimality results about the unique solvability for such boundary value problems are established.     

On then-th order iterative ordinary linear differential equations

Summary It is shown that a change of the basis in the solution space of a second order linear differential equation induces a covariant change in the solution space of the corresponding iterative equation. Also studied is the problem to what extent a solution of an iterative equation determines the equation.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

On oscillatory solutions of fourth order ordinary differential equations

The paper deals with the oscillation of a differential equation L ₄ y + P(t)L ₂ y + Q(t)y 0 as well as with the structure of its fundamental system of solutions.     

On global transformations of functional-differential equations of the first order

The paper describes the general form of functional-differential equations of the first order with m(m ≥ 1) delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation $f(t,uv,u_1 v_{\text{1}} , \ldots ,u_m v_m ) = f(x,v,v, \ldots ,v_m )g(t,x,u,u, \ldots ,u_m )u + h(t{\text{,}}x{\text{,}}u{\text{,}}u_{\text{1}} \ldots {\text{,}}u_m )v$ for u ≠ 0 is solved on $\mathbb{R}$ and a method of proof by J. Aczél is applied.     

Boundedness and stability criteria for linear ordinary differential equations of the second order

We establish some correlations for solutions of ordinary differential equations and the imaginary part of the complex solution of the corresponding Riccati equation. On the basis of these correlations and the I. M. Sobol’ theorem we prove some new stability and boundedness criteria for linear equations of the second order.     

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

Lie point symmetries for systems of second order linear ordinary differential equations

Abstract

The equations for the Lie point symmetries of autonomous systems of second order linear ordinary differential equations are derived. The results for a two dimensional system are treated in detail and some consideration is extended to higher dimensional systems. The effect of the introduction of time-dependent elements into the coefficient matrix is briefly discussed.