ИСТИНА

We consider system of differential equations of the form \begin{equation}\label{MainDifferentialEquation} \begin{pmatrix} y_1' \\ y_2' \end{pmatrix} = \lambda \begin{pmatrix} g & 0 \\ 0 & -h \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} +\begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{equation} where $g$ and $h$ are arbitrary functions such that $g,h > 0$ and $g, h, p, q, r, s \in W_1^k[0, 1]$, $k=0,1, \dots (W_1^0 = L_1)$. Under such requirements to coefficients we obtain asymptotics of fundamental system in half-plane $\{\Re{\lambda} > -d\}$ with an accuracy up to $o(\lambda^{-k})$, $\lambda \to \infty$. Using these results we show that non-regular problem \eqref{MainDifferentialEquation} with boundary conditions \begin{equation}\label{BoundaryConditions} y_1(0) = 0, \,\ y_2(1)=0 \end{equation} can be solved in similar manner to regular case with some modifications. The problem \eqref{MainDifferentialEquation}-\eqref{BoundaryConditions} is of interest in connection with applications in metallurgy.