On the normal systems of differential equations
On those ordinary differential equations that are solved exactly by the improved Euler method
As a numerical method for solving ordinary differential equations , the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method....
Parallelisms between differential and difference equations
The paper deals with the higher-order ordinary differential equations and the analogous higher-order difference equations and compares the corresponding fundamental concepts. Important dissimilarities appear for the moving frame method.
Particular solutions for a class of ODE related to the -exponential functions.
Polynomial algebra of constants of the Lotka-Volterra system
Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form , called the Lotka-Volterra derivation, where A,B,C ∈ k.
Polynomial solutions of a generalization of the first Painlevé differential equation.
Positive solutions for singular three-point boundary-value problems.
Power series techniques for a special Schrödinger operator and related difference equations.
Poznámka k integrování některých differenciálních rovnic lineárních
"Properties Of Solutions Of The Differential Equation x x""-k x' ²+f(x)=0"
Properties of solutions of the differential equation xx'' - kx'...+ f(x) = 0.
Quasi-exact solvability of a hyperbolic intermolecular potential induced by an effective mass step.
Quelques formules fondamentales pour l'étude des équations différentielles algébriques du premier ordre et du second degré entre deux variables et à intégrale générale algébrique
Recent applications of the theory of Lie systems in Ermakov systems.
Reduction of differential equations
Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) , where , and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of...
Remodified Bessel functions via coincidences and near coincidences.
Reproductivity of some equations of analysis. II.
Resolvent kernel for the Kohn Laplacian on Heisenberg groups.
Riccati equations.
Riccati's differential equation in birth-death processes.
This note reviews the occurrence of Riccati's equation in three birth-death type processes, and outlines their solutions.