Consider a system of ordinary differential equations dx/dt = X(x,y),dy/dt =Y(x,y),(1) where X(x, y) and Y(x,y) possess partial derivatives of first and second orders in a closed region S. To approach the system (1) we take the following system dx/dt=X_n(x,y),dy/dt=Y_n(x,,y),(2) where X_n(x,y) and Y_n(x,y) are polynomials of degree n. In order to determine X_n(x,y) and Y_n(x,y), two conditions will be imposed on. We call the first one the condition of expressibility, i.e. And the second one is the condition of approximation, the functional assuming the absolute minimum under the condition (3).The following general results are proved:(i) The necessary and sufficient condition that such pair of functions X_n(x, y) and Y_n(x, y) uniquely exist is that(ii) If such pair of functions uniquely exist for some non-negative integer n, it is also true for all integers m > n.(iii) Such pair of functions always uniquely exist either for the region S or for a suitable slight deformation of S. (iv) Divide S into subregions {S_i}. Let △ denote the maximal diameter of S(?)s. Let x_n(S_i) and Y_n(S_i) denote the uniquely determined functions for S_i. We haveConsider the case n=1. The integral curves of (2) form a system of conic section. Consider the neighbourhood of any point (X_o, y_o). Take S as a square with its center at the point (X_o, y_o) and its sides parallel with coordinate axes. Let the length of the side of the square shrink to zero, then we get a limiting conic section. The point (X_o, y_o) will then be called elliptic, hyperbolic and parabolic according as the limiting conic section being ellipse, hyperbola and parabola respectively. A region consisting of points of the same type will be called elliptic, hyperbolic and parabolic accordingly.The structure of integral curves in an elliptic region is very simple. The sign of the curvature of the curves remains unchanged. Hence the possible types of singularities are determined.The parabolic points either fill the whole region S or form one-dimensional curses in general. These curves are defined by the equation and separate the elliptic and hyperbolic regions.The structure of integral curves in a hyperbolic region is more complicated in general, and the region should be subdivided.Dividing S into regions of different types we can understand the structure of the integral curves much better. Examples are given to show how this method is applied to various problems.
