Solution of the equation d/dx(pdu/dx)+qu=cu by a solution of the equation d/dx(pdu/dx)+qu=0
Vladislav V. Kravchenko
TL;DR
The paper solves $\partial_{x}(p(x)\partial_{x}u(x)) + q(x)u(x) = \omega^{2}u(x)$ by leveraging a known solution of the homogeneous equation and a pseudoanalytic framework that links to the stationary Schrödinger equation. It develops a constructive method based on the Vekua equation and formal powers to build a pair of linearly independent solutions $u_1$ and $u_2$, valid for complex and real data, and shows how to extend to general $p$ via a factorization. Key contributions include explicit series representations in terms of $X^{(n)}$ and $\widetilde{X}^{(n)}$, independence of these coefficients from $\omega$, and connections to spectral problems and the Darboux transformation. The approach yields a numerically favorable procedure for a wide class of boundary-value and spectral problems in mathematical physics, enabling rapid computation of solutions for arbitrary $\omega$ once the base coefficients are prepared.
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.