The approximation method for approximating solutions of linear differential equations by algebraic polynomials

In this article we propose an effective method for the approximate solution of linear differential equations with polynomial coefficients, for which the coefficient $a_0(x)$ of the highest derivative is different from zero on the segment being considered. The approximating apparatus for the required solution $y(x)$ is provided by a certain sequence of polynomials $y_n(x)$. We prove that for $a_0(x)=\mathrm{const}$ the polynomials so constructed realize the asymptotically best approximation to the function $y(x)$ in the $L^2$ metric with Chebyshev weight, and that in the general case they have the property that
$$ \|y(x)-y_n(x)\|_C\leqslant AE_n(y)_C,\qquad E_n(y)_C=\inf_{c_k}\biggl\|y(x)-\sum_0^nc_kx^k\biggr\|,\quad A=\mathrm{const}. $$