Intégration qualitative des équations différentielles
In most cases we cannot integrate a differential equation in finite terms, using already known functions, for example using known combinations of elementary functions, or using functions defined by quadratures on such functions. If one wanted to restrict oneself to the cases where such an integration would be possible, the field of research on differential equations would be very limited and the great majority of the problems which arise in applications would be condemned to remain forever outside the field of our investigations.
"It is therefore necessary to study the functions defined by the differential equations in themselves and without trying to reduce them to simpler functions, as we have done for the algebraic functions which we had sought to reduce to radicals and that we are now studying directly, as has been done for the integrals of algebraic differentials, which for a long time we have attempted to express in finite terms." (H Poincaré)
In the theory of differential equations we have already taken a step in this direction by studying directly the integral in the neighbourhood of a point of the plane. But in these investigations, on the one hand, where one has taken the point of view of the general theory of functions, an equal importance is attached to the real and the imaginary parts; on the other hand, we limit ourselves to the study of the integral in the neighbourhood of a considered point. Now, in a host of important questions of Analysis, Geometry, Mechanics, Physics, etc., it is essential to know the integral in the real field. This study then has two parts:
- Qualitative part or the study of the general shape of the integral curve and the delimitation of a region of the plane comprising the curve passing through a given point, in part or in all its length;
- Quantitative part or numerical calculation of the ordinate of the curve corresponding to the given abscissa.
It is naturally by the qualitative part that must approach the study of the integral curve in the real field. In short, this study is reduced to that of the shape of the curve. We seek to construct this curve, that is to say one seeks its closed branches, its infinite branches, its asymptotes, the points of intersection with the coordinate axes, or with straight lines or fixed curves; the regions of the plane where the curve becomes imaginary; we examine whether the curve is oscillating, or constantly increasing or decreasing in a region (R) of the plane; if the oscillations are damped or increasing; the number and the position of the zeros are determined: maxima and minima, singular points in a region (R), the frequency of the oscillations, the speed of their damping, etc. We also try to frame the integral curve in part or in all its length, the problem is reduced to that of finding two fixed curves between which is included the arc of the integral curve when the abscissa or the ordinate vary between the given limits.
It is moreover clear that such a qualitative study, sufficiently advanced, is of the greatest utility for the quantitative determination of the integral curve itself. First, it facilitates the exact determination of a number of points of the curve. For this very reason it also facilitates the numerical calculation of the integral, because we already know convergent series which represent the integral in a region of the plane, and the main difficulty which presents itself as the extension of the numerical computation outside of this region, is to find a guide facilitating the passage from one region where the function is represented by a series, to another region where it is expressed by a different series. Considerations of this kind have indeed guided Henri Poincaré in these profound researches relating to the numerical computation of the integral of differential equations by means of series.
But the qualitative study also presents, by itself, an interest of the first order. Indeed, in applications, certain particularities such as zeros, infinites, minima, maxima, oscillating character, asymptotes of the curve which represent the appearance of a mechanical phenomenon, physical phenomenon, etc., are often what is most important to know. In the three body problem, for example, it is important to know whether one of the bodies will always remain in a certain region of the sky or whether it will be able to move away indefinitely; if the distance between two bodies will increase or decrease in the infinite limit, or if it will remain between certain fixed limits. In the question of the invariability of the elements of the planets, to show that the major axis has no secular variations, is to show that it oscillates between certain limits. In the study of the discharge of conductors with constant or variable capacitance, resistance and coefficient of self-induction, it is important in the first place to know whether the discharge is continuous or oscillating, to know the frequency of the oscillations and the direction in which it changes when these factors vary from one phenomenon to another, or during the same phenomenon.
These are real questions of qualitative analysis, not requiring the explicit knowledge of the integral equations to which the problem is reduced. The qualitative integration of the differential equations, providing the solution to questions of this nature, offers a vast field of research which it is impossible at present to traverse in all its extent and which will not be exhausted any time soon. We propose in this work to present briefly a set of currently known results, to the extent that space will allow.
The survey has four parts. In the first two we deal with the elements contributing to knowing the general aspect of the integral curve, providing data for the construction of this curve in outline. The third part deals with the oscillating integrals of the differential equations whose geometries have been particularly studied and whose theory contains a large number of acquired results. Finally, in the fourth part we will indicate various methods to frame a portion or a branch of the integral curve; assigning it as lower and upper limits of known curves.