Geodesic
The approximation method for approximating solutions of linear differential equations by algebraic polynomials
Izvestiya. Mathematics, Tome 8 (1974) no. 4, pp. 937-966 
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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     title = {The approximation method for approximating solutions of linear differential equations by algebraic polynomials},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $$ \|y(x)-y_n(x)\|_C\leqslant AE_n(y)_C,\qquad E_n(y)_C=\inf_{c_k}\biggl\|y(x)-\sum_0^nc_kx^k\biggr\|,\quad A=\mathrm{const}. $$

[1] Hart J. F., Cheney E. W., Lawson C. L., Maehly H. J., Mesztenyi C. K., Rice J. R., Thacher Jr. H. G., Witzgall C., Computer approximations, New York, London, Sydney, 1968

[2] Bernshtein S. N., Ekstremalnye svoistva polinomov i nailuchshee priblizhenie nepreryvnykh funktsii odnoi veschestvennoi peremennoi, ONTI, M., L., 1937

[3] Dzyadyk V. K., “O primenenii lineinykh metodov k priblizheniyu polinomami reshenii obyknovennykh differentsialnykh uravnenii i integralnykh uravnenii Gammershteina”, Izv. AN SSSR. Ser. matem., 34 (1970), 827–848 | Zbl

[4] Dzyadyk V. K., “Ob effektivnom postroenii mnogochlenov, kotorye osuschestvlyayut blizkoe k nailuchshemu priblizhenie funktsii $e^x$, $\sin x$ i dr.”, Ukr. matem. zh., 25:4 (1973), 435–453 | MR

[5] Trikomi F., Integralnye uravneniya, IL, M., 1960

[6] Stolyarchuk V. K., “O ravnomernom priblizhenii mnogochlenami na segmente funktsii Besselya s tselymi indeksami”, Ukr. matem. zh., 26:5 (1974)

[7] Bekkenbakh E., Bellman R., Neravenstva, Mir, M., 1965 | MR

[8] Uitteker G. T., Vatson G. N., Kurs sovremennogo analiza, I, II, GTTI, M.