Solution of the equation d/dx(pdu/dx)+qu=cu by a solution of the equation d/dx(pdu/dx)+qu=0

TL;DR

This paper presents a straightforward method to solve the inhomogeneous differential equation d/dx(pdu/dx)+qu=cu using known solutions of its homogeneous counterpart, leveraging pseudoanalytic function theory.

Contribution

It introduces a simple, verifiable formula for the general solution based on the theory of pseudoanalytic functions and their connection to the stationary Schrödinger equation.

Findings

Derived a simple solution formula for the inhomogeneous equation

Connected solutions of the differential equation to pseudoanalytic functions

Provided an easily applicable method for solving similar equations

Abstract

We give a simple solution of the equation d/dx(pdu/dx)+qu=cu whenever a nontrivial solution of d/dx(pdu/dx)+qu=0 is known. The method developed for obtaining this result is based on the theory of pseudoanalytic functions and their relationship with solutions of the stationary two-dimensional Schrodinger equation. The final result, that is the formula for the general solution of the equation d/dx(pdu/dx)+qu=cu has a simple and easily verifiable form.