On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

Nonclassical behavior of solutions of linear second-order ordinary differential equations

We present a number of unstable second-order equations of the form

$$y'' + (1 + g(x))y = 0,$$

where the coefficient g(x) satisfies the conditions g(x) ∈ C(0, ∞) and lim_x→+∞ g(x)=0 but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials).

     

Complete group classification of systems of two linear second-order ordinary differential equations

We give a complete group classification of the general case of linear systems of two second-order ordinary differential equations excluding the case of systems which are studied in the literature. This paper gives the initial step in the study of nonlinear systems of two second-order ordinary differential equations. It can also be extended to systems of equations with more than two equations. Furthermore the complete group classification of a system of two linear second-order ordinary differential equations is done. Four cases of linear systems of equations with inconstant coefficients are obtained.     

Periodic solutions of systems of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives

We establish sufficient conditions for the existence of a periodic solution of a system of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives in the case of an arbitrary periodic inhomogeneity. Ternopol Academy of National Economy, Ternopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 350–356, March, 1998.     

Symmetries and invariants of the systems of two linear second-order ordinary differential equations

For arbitrary systems of two linear second-order ordinary differential equations, the symmetry Lie algebra is described in terms of invariant theory, resulting in eleven non-equivalent symmetry types. The result is compared with the group classification approach recently obtained by different authors.     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

On two comparison tests for linear second-order ordinary differential equations

We prove two comparison tests for second-order linear differential equations. The well-known Sturm comparison theorem is a straightforward corollary of the first of them. The second (integral) test permits one to use specific integral relations between the coefficients of two equations to prove the oscillation of one of the equations assuming the oscillation of the other.     

A procedure for solving some second-order linear ordinary differential equations

The purpose of this work is to introduce the concept of pseudo-exactness for second-order linear ordinary differential equations (ODEs), and then to try to solve some specific ODEs.     

To the theory of degenerate systems of ordinary differential equations

Translated fromSibirskii Matemaiicheskii Zhurnal, Vol. 36, No. 5, pp. 1194–1207, September–October, 1995.     

Asymptotic behavior of solutions of linear ordinary differential equations

We present some conditions which ensure that the solution Y(x) of the ordinary differential equation Y′(x) = A(x) Y(x), Y(x₀) = I, where x₀ ? x < ∞ and A(x), Y(x) are n × n complex matrix-valued functions with A(x) continuous, has a nonsingular limit as x → ∞.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

Bounded solutions of second-order systems of differential equations

Sufficient conditions for the existence, uniqueness and stability of the bounded (in ) solution of the second-order systems of nonlinear differential equations are given.     

Positive solutions for a multi-parameter system of second-order ordinary differential equations

In this paper, we establish the product formula for the fixed point index on product cone, andthen, as applications, consider the existence, nonexistence and multiplicity of positive solutions for a second-order ordinary differential system with parameters. The discussion is based on the product formula and thefundamental properties of the fixed point index.