Abstract
This paper aims to use some non-standard concepts to find a non-standard analytic and non-analytic infinitely close solution of the first-order ordinary differential equation in the monad of its singularity, where the differential coefficients are either infinitesimal, unlimited or have basic differential form. The obtained non-standard solutions are more precise and compatible than the conventional ones. We named such a non-analytic infinitely close solution to the singularity by shadow solution. These cases of solutions are sometimes impossible to obtain by conventional methods.
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