Related papers: Integrability of Riccati equation from a group the…
Several instances of integrable Riccati equations are analyzed from the geometric perspective of the theory of Lie systems. This provides us a unifying viewpoint for previous approaches.
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides…
Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei-Norman method is applied to obtain the associated…
A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytic general solution is presented,…
In this Chapter, using Riccati equation as our main example, we tried to demonstrate at least some of the ideas and notions introduced in Chapter 1 - integrability in quadratures, conservation laws, etc. Regarding transformation group and…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…
It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also…
The geometric theory of Lie systems will be used to establish integrability conditions for several systems of differential equations, in particular Riccati equations and Ermakov systems. Many different integrability criteria in the…
A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if…
A general Riccati equation is integrated in quadratures in case one of its coefficients is an arbitrary function and two others are expressed through it.
An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.
The Riccati equation method is used to establish a new comparison theorem for systems of two linear first order ordinary differential equation. This result is based on a, so called, concept of "null-classes", and is a generalization of…
The Riccati equation method and an approach of the use of unknown factors is used to establish oscillation, suboscillation and nonoscillation criteria for linear systems of ordinary differential equations. A necessary condition for Lyapunov…
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler…
A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead…
The connection of unbroken SUSY quantum mechanics in its strictly isospectral form with the nonlinear Riccati superposition principle is pointed out
An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati…
Ten new exact solutions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The solutions are obtained by assuming certain relations among the coefficients $a(x)$, $b(x)$ and $c(x)$ of the Riccati equation, in the form of…