Majorana Transformation for Differential Equations


We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of nonlinearity of the lower order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden–Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas–Fermi equation.

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  1. Bellman, R. (1953). Stability Theory forDifferential Equations, McGraw-Hill, New York.

    Google Scholar 

  2. Bender, C.M. and Orszag, S. A. (1978). Advanced MathematicalMethods for Scientists and Engineers, McGraw-Hill, New York.

    Google Scholar 

  3. Esposito, S. (2002). American Journal of Physics 70, 852.

    Google Scholar 

  4. Esposito, S., Majorana, E., Jr., van der Merwe, A., and Recami, E. (inpress). Ettore Majorana: Notebooks in Theoretical Physics, Kluwer, New York.

  5. Polyamin, A. D. and Zaitsev, V. F. (1995). Handbook of ExactSolutions for Ordinary Differential Equations, CRC, New York.

    Google Scholar 

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Esposito, S. Majorana Transformation for Differential Equations. International Journal of Theoretical Physics 41, 2417–2426 (2002).

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  • Majorana scale-invariant differential equations
  • Emden–Fowler equations
  • solution of the Thomas–Fermi equation