The classification of differential equations follows from one single question: can we  calculate formally the solution if sufficiently many initial data are given? Consider the initial problem for an ordinary differential equation $$y'(x)=f(x,y(x))$$, $$y(x_0)=y_0$$. Then one can determine formally the solution, provided the function $$f(x,y)$$ is sufficiently regular. The solution of the initial value problem is formally given by a power series. This formal solution is a solution of the problem if $$f(x,y)$$ is real analytic according to a theorem of Cauchy. In the case of partial differential equations the related theorem is the Theorem of Cauchy-Kowalevskaya. Even in the case of ordinary differential equations the situation is more complicated if $$y'$$ is implicitly defined, i. e., the differential equation is $$F(x,y(x),y'(x))=0$$ for a given function $$F$$.