# Spectral parameter power series representation for solutions of linear system of two first order differential equations

## TL;DR

This paper introduces a spectral parameter power series (SPPS) method for solving linear systems of two first-order differential equations, providing a recursive integral-based series representation for solutions with broad applicability.

## Contribution

The paper develops a new SPPS representation for solutions of first-order linear systems with arbitrary matrix coefficients, including a transformation for more general systems and numerical implementation guidance.

## Key findings

- SPPS series converges and exists for the systems considered.
- Transformation reduces general systems to the form suitable for SPPS.
- Numerical examples demonstrate the effectiveness of the method.

## Abstract

A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter, \[ B \frac{dY}{dx} + P(x)Y = \lambda R(x)Y,\] where $Y=(y_1,y_2)^T$ is the unknown vector-function, $\lambda$ is the spectral parameter, $B = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, and $P$ is a symmetric $2\times 2$ matrix, $R$ is an arbitrary $2\times 2$ matrix whose entries are integrable complex-valued functions. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a non-vanishing solution for one particular $\lambda = \lambda_0$. The existence of such solution is shown. For a general linear system of two first order differential equations \[ P(x)\frac{dY}{dx}+Q(x)Y = \lambda R(x)Y,\ x\in [a,b], \] where $P$, $Q$, $R$ are $2\times 2$ matrices whose entries are integrable complex-valued functions, $P$ being invertible for every $x$, a transformation reducing it to a type considered above is shown. The general scheme of application of the SPPS representation to the solution of initial value and spectral problems as well as numerical illustrations are provided.