Reduction of underdetermined systems of ordinary differential equations: I

We consider the equivalence problem for underdetermined systems of ordinary differential equations. We present canonical forms for some types of autonomous systems linear in the derivatives. It is shown that, among three-dimensional autonomous systems linear in the derivatives, there are infinitely many locally nonequivalent systems.     

Reduction of underdetermined systems of ordinary differential equations: II

We study the equivalence of underdetermined systems of ordinary differential equations and present a local classification of systems that are linear in derivatives, are sufficiently regular, and consist of two equations.     

Reduction of underdetermined systems of ordinary differential equations: III

We consider the problem on the restriction of underdetermined systems of ordinary differential equations linear in the derivatives to a manifold.     

Reduction of underdetermined systems of ordinary differential equations: IV

We study the problem on the decomposition of underdetermined systems of ordinary differential equations linear in the derivatives. Decomposition permits one to reduce the construction of solutions of the original system to the successive construction of solutions of systems that contain fewer unknowns and equations.     

On certain types of point symmetries of systems of second-order ordinary differential equations

Existence criteria for some generic types of point symmetries of systems of n-second order ordinary differential equations are studied, specially in connection with the generation of semisimple subalgebras of symmetries belonging to the simple linear and orthogonal types, as well as their maximal dimension and rank. The structure of certain time-dependent symmetries, in particular scaling symmetries, are also studied, and the structure of the subalgebras they span determined. Generic examples illustrating the procedure are given.     

On the stability of solutions of certain systems of ordinary differential equations

Summary The object of the paper is to give n-dimensional extensions of some stability results for certain Aizerman-type systems of differential equations of order two.     

Solution by quadratures of certain ordinary differential equations

This paper discusses methods that are applicable in the solution by quadratures of linear second‐order differential equations with variable coefficients. These same techniques, when applied to equations with constant coefficients, produce an extremely useful method in the teachingof ordinary differential equations.     

Flag systems and ordinary differential equations

We discuss the Monge problem in the theory of ordinary differential equations and prove the Cartan criterion for first order Monge systems with the added hypothesis of homogeneity. Next, we examine Hilbert's counter-example and finally give a brief account on the two special cases involving flag systems of length two and three. Entrata in Redazione il 1 agosto 1998.     

Heteroclinic period blow-up in certain symmetric ordinary differential equations

We consider the problem of bifurcation as well as of accumulation of periodic orbits on heteroclinic orbits for certain systems of ordinary differential equations either equivariant under finite groups of linear transformations or periodic in spatial variables.     

On comparison systems for ordinary differential equations

We state and prove general comparison principles for ordinary differential equations, discuss the scope of these principles, and present examples.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Convergence of solutions of certain ordinary third order differential equations

Summary This paper is sequel to an earlier one [2]. Sufficient conditions are established for solutions of to converge. Entrata in Redazione il 9 ottobre 1971.     

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.     

On oscillatory solutions of certain first order ordinary differential equations

By constructing a class of solutions to the integral inequality for t  t₀ large enough, where 0<A₁a(τ)A₂<+∞ and λ>1, that tend to zero as t→+∞ we address an open problem in the theory of nonlinear oscillations.     

Boundary-value problems for systems of ordinary differential equations

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3–103, 1987.     

A useful result for certain linear periodic ordinary differential equations

In this paper we derive some properties for systems of linear ordinary differential equations with time-periodic coefficient matrix satisfying an additional symmetry condition (Hale's property E). These results can be used in the numerical integration of such systems and reduce the computer time by 50 %.