A Novel Method to Solve Familiar Differential Equations and its Applications

Abstract

Text books on mathematical methods [1] and non-relativistic quantum mechanics [2] routinely take recourse to the method of series solution for solving linear differential equations encountered in various physical problems. The tediousness of the above procedure has often led to the search for alternate methods; the elegant raising and lowering operator approach for solving the harmonic oscillator problem serves as a classic example of these attempts. The other algebraic approach is group theoretical in nature and takes advantage of the symmetries of the problem at hand [2, 3]. The text book example of the Coulomb problem, where the angular momentum and the Rungé-Lenz vectors combine to yield a complete algebraic description of the Hilbert space, demonstrates the efficacy of this procedure. Most of these methods fail to generalize to many-variable interacting systems, a field lately attracting considerable attention because of its relevance to many areas of physics [4, 5].