# On Lie systems and Kummer-Schwarz equations

## Related Articles

- Oscillation on a Class of Differential Equations of Fractional Order. Tongbo Liu; Bin Zheng; Fanwei Meng // Mathematical Problems in Engineering;2013, p1
Based on Riccati transformation and certain inequality technique, some new oscillatory criteria are established for the solutions of a class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The oscillatory criteria established are...

- Movable algebraic singularities of second-order ordinary differential equations. Filipuk, G.; Halburd, R. G. // Journal of Mathematical Physics;Feb2009, Vol. 50 Issue 2, pN.PAG
Any nonlinear equation of the form yâ€³=âˆ‘n=0Nan(z)yn has a solution with leading behavior proportional to (z-z0)-2/(N-1) about a point z0, where the coefficients an are analytic at z0 and aN(z0)â‰ 0. Equations are considered for which each possible leading term of this form extends...

- HYPERGEOMETRIC SOLUTIONS TO AN ULTRADISCRETE PAINLEVÃ‰ EQUATION. Ormerod, Christopher M. // Journal of Nonlinear Mathematical Physics (World Scientific Publ;Feb2010, p87
We show that an ultradiscrete analogue of the third PainlevÃ© equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework...

- Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives. Shouxian Xiang; Zhenlai Han; Ping Zhao; Ying Sun // Abstract & Applied Analysis;2014, p1
By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation [a(t)(p(t) + q(t)(D_Î±)(t))Î³]' - b(t)f(âˆ«tâˆž(s - t)-Î± x(s)ds) = 0, for t â‰¥ t0 0, where D_Î±x is the Liouville...

- Expansion of solutions of the sixth PainlevÃ© equation near a regular point. Goryuchkina, I. // Journal of Mathematical Sciences;Sep2007, Vol. 145 Issue 5, p5173
For the sixth PainlevÃ© equation, by using power-geometry methods, all asymptotic expansions of solutions are obtained in a neighborhood of a regular point of the independent variable. All of these expansions are convergent series in integer powers with constant complex coefficients. Five...

- HYPERGEOMETRIC SOLUTIONS TO AN ULTRADISCRETE PAINLEVÃ‰ EQUATION. Ormerod, Christopher M. // Journal of Nonlinear Mathematical Physics (World Scientific Publ;Mar2010, Vol. 17 Issue 1, p87
We show that an ultradiscrete analogue of the third PainlevÃ© equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework...

- Auto-BÃ¤cklund transformations for a differential-delay equation. Gordoa, Pilar R.; Pickering, Andrew // Journal of Mathematical Physics;Mar2013, Vol. 54 Issue 3, p033502
Discrete PainlevÃ© equations have, over recent years, generated much interest. One property of such equations that is considered to be particularly important is the existence of auto-BÃ¤cklund transformations, that is, mappings between solutions of the equation in question, usually involving...

- Asymptotic properties of solutions of Riccati matrix equations and inequalities for discrete symplectic systems. Hilscher, Roman Šimon // Electronic Journal of Qualitative Theory of Differential Equatio;2015, Issue 48-54, p1
In this paper we study the asymptotic properties of the distinguished solutions of Riccati matrix equations and inequalities for discrete symplectic systems. In particular, we generalize the inequalities known for symmetric solutions of Riccati matrix equations to Riccati matrix inequalities. We...

- A GEOMETRIC APPROACH TO INTEGRABILITY CONDITIONS FOR RICCATI EQUATIONS. Cariñena, José F.; De Lucas, Javier; Ramos, Arturo // Electronic Journal of Differential Equations;2007, Vol. 2007, p1
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.