Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1268–1281 | Cite as

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

  • S. CarilloEmail author
  • F. Zullo


We study the class of nonlinear ordinary differential equations y″ y = F(z, y2), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.


nonlinear ordinary differential equation Bäcklund transformation Schwarzian derivative Ermakov–Pinney equation Emden–Fowler equation 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze di Base e Applicate per l’IngegneriaUniversità di Roma “La Sapienza,”RomeItaly
  2. 2.Instituto Nazionale di Fisica NucleareRomeItaly
  3. 3.DICATAMUniversità di BresciaBresciaItaly

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