Exact analytic solutions of the Abel, Emden–Fowler and generalized Emden–Fowler nonlinear ODEs

Abstract

Several basic particular nonlinear ordinary differential equations (ODEs) of the second-order in mathematical physics and nonlinear mechanics are reduced to equivalent equations of the Abel normal form yy x ′ - y = f ( x ) by means of various admissible functional transformations. These equivalent equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of the above type of Abel equation can be solved in parametric form [Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen, vol. 1, B.G. Teubner, Stuttgard, 1977; Polyanin and Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, New York, 1999]. In this paper, a successful attempt is made to present a mathematical construction leading to the exact analytic solution of the above Abel equation. Since there are admissible functional transformations that reduce the Emden–Fowler equation y xx ″ = Ax n y m and the generalized Emden–Fowler equation y xx ″ = Ax n y m ( y x ′ ) ℓ to the above Abel equation, the developed construction concerns also the analytic solutions of these two types of Emden–Fowler's nonlinear ODEs.