We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel's nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one-parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel's equation of the first kind.

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Two kinds of second-order nonlinear, ordinary differential equations (ODEs) appearing in mathematical physics are analyzed in this paper. The first one concerns the Thomas-Fermi (TF) equation, while the second concerns the Langmuir-Blodgett (LB) equation in current flow. According to a mathematical methodology recently developed, the exact analytic solutions of both TF and LB ODEs are proposed. Both of these are nonlinear of the second order and by a series of admissible functional transformations are reduced to Abel’s equations of the second kind of the normal form. The closed form solutions of the TF and LB equations in the phase and physical plane are given. Finally a new interesting result has been obtained related to the derivative of the TF function at the limit.

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We present a mathematical methodology for constructing the exact parametric solution of the first Painlevé second order nonlinear ordinary differential equation. Several admissible functional transformations are introduced through an intermediary analysis delivering us from the a priori construction of power series solutions..

Η τάξη των εξισώσεων Ε. Kamke (1976) M. Murphy (1960) A.D. Polyanin και V.F. Zaitsev (1999) ∆εύτερη τάξη 249 315 1227 Τρίτη τάξη 13 22 587 Τέταρτη τάξη 3 3 75 Μεγαλύτερη τάξη 3 9 160 Συνολικός Αριθµός Εξισώσεων 268 349 2049 Κατά τη συγκέντρωση του υλικού κυρίως στο τρίτο των παραπάνω συγγραµµάτων, οι συγγραφείς έδωσαν προβάδισµα σε δυο τύπους εξισώσεων: 13.