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This work presents a direct and highly accurate method to solve ordinarydifferential equations, in particular the Schr\"odinger equation in onedimension, through the direct substitution of a power series solution to obtaina purely algebraical system containing the recurrence relations among theseries coefficients. With these recurrence relations at hand it is possible tobuild an extremely simple routine using only basic arithmetic operations tofind solutions of very high accuracy at a very low cost of machine resources.For the Schr\"odinger equation the energy eigenvalues may be estimated by avery simple method, an this estimate may be easily refined using the recurrencerelations until the specified tolerance has been reached, which allows one tofind high accuracy wavefunctions even for states with very high quantumnumbers.    Este trabalho apresenta um m\'etodo direto e de alta acur\'acia para resolverequa\c{c}\~oes diferenciais ordin\'arias, em particular a equa\c{c}\~ao deSchr\"odinger em uma dimens\~ao, atrav\'es da substitui\c{c}\~ao direta de umasolu\c{c}\~ao em s\'erie de pot\^encias para obter um sistema puramentealg\'ebrico contendo as rela\c{c}\~oes de recorr\^encia entre os coeficientesda s\'erie. De posse dessas rela\c{c}\~oes de recorr\^encia \'e poss\'ivelconstruir uma rotina extremamente simples, usando somente opera\c{c}\~oesaritm\'eticas b\'asicas, para encontrar solu\c{c}\~oes de alt\'issimaacur\'acia a um custo muito reduzido de recursos de m\'aquina. Para aequa\c{c}\~ao de Schr\"odinger, os autovalores de energia podem ser estimadospor um m\'etodo simples, e a estimativa pode ser facilmente refinada com o usodas rela\c{c}\~oes de recorr\^encia at\'e que se alcance a toler\^anciaespecificada, o que permite encontrar fun\c{c}\~oes de onda de alta acur\'aciamesmo para estados com n\'umero qu\^antico alto.

Resolvendo a equa\c{c}\~ao de Schr\"odinger atrav\'es da substitui\c{c}\~ao direta da s\'erie de pot\^encias