||The classical period of development of the mathematical analysis of the 18TH century - has left a legacy of mathematics of the so-called elementary methods of integration of differential equations; then the same was mainly highlighted the class of equations, in which finding the general solution is reduced to quadratures or algebraic operations. The first half of the 19TH century is marked by the critics of this inheritance in two directions. On the one hand, Кошн places and for a wide class of equations allows the problem of existence of a solution. On the other hand, Лиувилль proves the impossibility of finding in quadratures of the general solution of the special Riccati Equation, except for the known cases, when this decision is expressed in the form of a combination of excellence and rational functions. This discovery significantly depreciated the finding of new cases of elementary integrability.
Existence theorems of opened theoretical path for the approximation and numerical methods, which, however, started to develop, regardless of any theory, even in the previous period, under the influence of the urgent demands of applied mathematics, in particular heaven-