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

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, $B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, 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.