On the Applications of a New Technique to Solve Linear Differential Equations, with and without Source

Abstract

A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equa- tions, both without and with a source term. A new quasi-solvable potential has also been constructed taking recourse to the above method. Linear differential equations play a crucial role in various branches of science and mathematics. Second order differential equations routinely manifest in the study of quantum mechanics, in connection with Schrodinger equation. There are various techniques available to solve a given differential equation, e.g., power series method, Laplace transforms, etc. Not many general methods applicable to differential equations of arbitrary order exist in the literature (see (1) and references therein). We make use of a general method for solving linear differential equations of arbitrary order to construct new representations for the solutions of the known second order linear differential equations, both without and with a source term. This method has found applications in solving linear single and multi variable differential equations. The solutions of linear differential equation with a source and development of a new Quasi-Exactly Soluble (QES) system are the new results of this paper.