Exact analytic solutions of the Abel, Emden-Fowler and generalized Emden-Fowler nonlinear ODEs
Eight particular so far unsolvable classical second order nonlinear ordinary differential equations (ODEs) in mathematical physics and nonlinear mechanics (the Rayleigh, Emden's, simplified Blasius, Duffing, Langmuir's, van der Pol, Thomas-Fermi and Kidder's equations, respectively) are reduced by several admissible functional transformations to an equivalent Abel's equation of the second kind of the normal form \(yy' - y = f(x)\). A mathematical technique leading to the construction of the exact analytic solutions concerning the above type of Abel equation is provided. The presented method can also be used for the Emden-Fowler equations \(y''_{xx} = A x^n y ^m\) and the generalized Emden-Fowler equations \(y''_{xx} = A x^n y^m (y'_x)^{\ell}\) because there exist admissible functional transformations [\textit{A. D. Polyanin, V. F. Zaitsev}, Handbook of Exact Solutions for Ordinary Differential Equations, (Boca Raton), FL: CRC Press, ( 1995; Zbl 0855.34001)]) that can reduce these equations to Abel's equation of the second kind of the normal form.
scientific article; zbMATH DE number 5116300
scientific article