On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Method for constructing solutions of linear ordinary differential equations with constant coefficients

For Cauchy problems involving linear differential equations with constant coefficients, a new method for constructing solutions without determining the roots of the characteristic equation is proposed. Formulas for the differentiation of the solution with respect to the equation coefficients are derived, and an approximate analytical solution is found.     

On the group classification of systems of two linear second-order ordinary differential equations with constant coefficients

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases correspond to the type of equations where the commutative property of the coefficient matrices with respect to the dependent variables and the first-order derivatives in the considered system does not hold. A discussion of the results as well as a note on the extension to linear systems of second-order ordinary differential equations with more than two equations are given.     

On the solutions of linear differential equations with singular coefficients

Equations of the form E? = Ax + u, with E and A square matrices and E singular, are considered. The controversy that exists in the literature concerning the solutions of such equations is investigated. Solutions are arrived at through an application of singular perturbation theory.     

On solutions of linear differential equations with entire coefficients

The first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions. Subsequent results involve the separation of the zeros of a Bank-Laine function.     

On the unique continuation along curves of germs of solutions to linear differential equations with constant coefficients

Mathematical Notes -     

New features of stability of linear systems of differential equations with constant coefficients

We consider stability of linear systems of differential equations with constant real coefficients whose matrices are off-diagonally non-negative. The results are applied to arbitrary linear systems of differential equations with constant complex coefficients.     

Pathology of infinite systems of linear first order differential equations with constant coefficients

Summary The properties of solutions of finite systems are analyzed and it is shown by examples how these properties may be distorted and ultimately lost in passing from finite to infinite systems. To Enrico Bompiani on his Scientific Jubilee This research was supported in part by United States Army through its Office of Ordnance Research under Grant DA-ORD-12. — Most of the results have been presented to various organisations such as Section A of the American Association for the Advancement of Science and mathematics colloquia at the universities of Leningrad, Moscow, and Toronto.     

On algebroid solutions of linear differential equations with algebroid coefficients

We establish a relationship between the growth rates of algebroid solutions and coefficients of nth-order linear differential equations.     

On the singularities of solutions of partial differential equations with constant coefficients

LetP be a differential operator with constant coefficients in ℝ ⁿ . Ifu is a distribution, the singular support ofu is the complement of the largest set whereu ∈C ^∞. Necessary and sufficient conditinos are obtained for a closed convex set Γ to be equal to the singular support ofu for someu withPu ∈C ^∞ or, equivalently, for Γ to contain the singular support ofu for someu withPu ∈C ^∞ butu ∉C ^∞. Related local uniqueness theorems analogous to the Holmgren theorem with supports replaced by singular supports are also given, as well as applications concerningP-convexity with respect to singular supports.     

Critical oscillation constant for half-linear differential equations with periodic coefficients

We compute explicitly the oscillation constant for certain half-linear second-order differential equations involving periodic coefficients. If these periodic functions are constants, our results reduce to the well-known oscillation constants for half-linear Euler and Riemann–Weber differential equations.     

On the stability of the linear differential equation of higher order with constant coefficients

We obtain a result on stability of the linear differential equation of higher order with constant coefficients in Aoki-Rassias sense. As a consequence we obtain the Hyers-Ulam stability of the above mentioned equation. A connection with dynamical sytems perturbation is established.