Some Nonstandard Treatment of the Singularity in the Differential Equation

This paper aims to use some nonstandard concepts to find a nonstandard 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 nonstandard 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