Closed-Form Solution of Second Order Linear Ordinary Differential Equations

Authors: Saeed Otarod

Abstract:

A transformational method is employed to obtain closed-form integral solutions for nonhomogeneous second order linear ordinary differential equations in terms of a particular solution of the corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is first transformed into a simple Riccati equation from which the general solution of the nonhomogeneous second order linear differential equation, in the form of a closed integral equation, is inferred. The method is applied to the solution of Schr¨odinger equation for hydrogen-like atoms. A generic nonhomogeneous second order linear differential equation has also been solved to further exemplify the methodology.

Keywords: Closed form, Second order ordinary differential equations, explicit, linear equations, differential equations.

References:

[1] E.L. Ince, “Ordinary Dierential Equations,” New York: Dover Publications, 1956.
[2] G.B. Arfken and H. J. Weber, “ Mathematical Methods for physicists,” 4th ed., Elsevere, 2013.
[3] C.R. Wylie , “Advanced Engineering Mathematics,” 4th ed., New York: McGraw-Hill, 1975.
[4] R. T. William, “Riccati Dierential Equations, London,” Academic Press, 1972.
[5] B.H. Bransden and C. J. Joachain, C. J., “Physics of atoms and molecules,” Prentice Hall, 2010.
[6] S. Otarod, “Summary of atomic quantum numbers,” Lecture note, 28 July 2006.
[7] S.I. Hayek, “Mechanical Vibration and Damping: Encyclopedia of Applied Physics,” WILEY-VCH Verlag GmbH and Co KGaA, Apr. 2003.