Geodesic
Ermakov–Pinney and Emden–Fowler equations: New solutions from novel Bäcklund transformations
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 373-389 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the class of nonlinear ordinary differential equations $y''y= F(z,y^2)$, 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.
Keywords: nonlinear ordinary differential equation, Bäcklund transformation, Schwarzian derivative, Ermakov–Pinney equation, Emden–Fowler equation. 
@article{TMF_2018_196_3_a1,
     author = {S. Carillo and F. Zullo},
     title = {Ermakov{\textendash}Pinney and {Emden{\textendash}Fowler} equations: {New} solutions from novel {B\"acklund} transformations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {373--389},
     year = {2018},
     volume = {196},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a1/}
}
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S. Carillo; F. Zullo. Ermakov–Pinney and Emden–Fowler equations: New solutions from novel Bäcklund transformations. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 373-389. http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a1/