Geodesic
The approximation method for approximating solutions of linear differential equations by algebraic polynomials
Izvestiya. Mathematics , Tome 8 (1974) no. 4, pp. 937-966.

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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 $$ \|y(x)-y_n(x)\|_C\leqslant AE_n(y)_C,\qquad E_n(y)_C=\inf_{c_k}\biggl\|y(x)-\sum_0^nc_kx^k\biggr\|,\quad A=\mathrm{const}. $$ 
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V. K. Dzyadyk. The approximation method for approximating solutions of linear differential equations by algebraic polynomials. Izvestiya. Mathematics , Tome 8 (1974) no. 4, pp. 937-966. http://geodesic.mathdoc.fr/item/IM2_1974_8_4_a7/

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