Error estimation for the third-order accuracy approximate solution of the Cauchy problem by the Taylor formula

A method based on the Taylor formula for the approximate solution of the Cauchy problem for the ordinary differential equation is studied. The problem of estimating the accuracy of the approximate solution generated by this method is considered, and an estimate of high accuracy is obtained for the difference of the exact and approximate solution. Here, different from the known values, an exact expression for the estimation coefficient is found.

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