EQUATIONS, Differential. 1. Introduc invention of the calculus, made necessary by the demands of natural science, was followed immediately by the most brilliant applications. The names of Newton, Leibnitz, Euler, Lagrange and Laplace are attached to the principal discoveries of this period, whose importance from a scientific and philosophical point of view can hardly be overestimated. A simple example will suffice to explain the ruling idea of this epoch. From the observ ations of Tycho Brahe, Kepler had obtained the laws of planetary motion still known by his name. Newton had shown that Kepler's laws were but a consequence of the laws of universal gravitation, which assumes that every particle in the universe acts upon every other according to a definite law. The effect of Newton's law upon a system of moving bodies can be formulated in mathematical symbols without any difficulty. This formulation gives rise to a system of equations involving the co ordinates of the moving bodies and their accelerations,. i.e., the second derivatives of these co-ordinates with respect to the time. The problem of expressing the co-ordinates as functions of the time, i.c., the problem of integrating this system of differential equations, was solved by Newton for the case of two mutually attracting bodies, and its solution is given precisely by Kepler's laws. Newton him self and his successors, especially Laplace and Lagrange, studied the further consequences of the law of gravitation as applied to the solar system. The accord between the theory and ob servation became closer and closer, so that it was reasonable to suppose that the true law of nature had been found. Gradually other branches of physical science were treated in a similar way. In all cases, the fundamental laws being assumed, the mathematical formula tion of the problems led to the question of integrating differential equations. It should be noted that, although in some cases this method of arriving at the formulation of the physical problems has now been abandoned, differential equations are now, more than ever, used as the expressions for the fundamental phenom ena in physical science. For the applications of mathematics there is no field so important as the theory of differential equations. That the whole world is a mathematical problem was the point of view gained by Laplace, an insight gained in a different way also by Leib nitz and Spinoza. But the mathematician is more specific; we learn from him that this world-problem belongs to the domain of the theory of differential equations. Even if the details of the picture have changed, the formu lation of this general idea is one of the posi tive achievements of the philosophical thought of the 18th century.
Ordinary Differential Equations; Elemen tary Let y be determined as a function of x by means of an equation, (1) 95(x, y, a)=0, which involves an arbitrary constant a. If x and y be interpreted as the co-ordinates of a point in the plane, equation (1) represents a family of curves, one curve for each value of a.
By differentiation we find, from (1), ao , a I) dy (2) -I- — — ---- O.
ax ay dx Between these two equations a may be elimi nated; the result will be an equation of the form dy\ (3) f free from a. Equation (3) is a differential equation. Since it does not contain the con stant a it gives the expression of a property which is common to all of the curves of the family (1). The main object of the theory of differential equations is to invert the proc ess which we have just carried out, i.e., the equation (3) being given, the equation(1) involving an arbitrary constant, from which (3) may be derived by differentiation, is to be found. This process is known as the integra tion of the differential equation.
In general let there be given an equation of the form dy f dx' ' ' • d &of =- between x, the function y of x and its deriva tives up to the nth order; it is called an ordi nary differential equation of the nth order. The adjective ordinary implies that y is considered as a function of only one independent variable x. Under certain restrictions as to the continuity of the function f (a question to which we shall recur later), it may be shown that there exists a function y of x and of n arbitrary constants which satisfies the differential equation ; it is known as the general integral of the differential equation; the determination of this function is the object of the theory of differential equations. The equation is then said to have been inte grated.
The simplest case of such a differential equa tion presented itself in the problem of finding the area included between a curve 3f (x), the x-axis, and two ordinates erected for x = a and x=x. The differential equation satisfied by the area is considered as function of x is ds and the area itself becomes f(x)dx. a This simple case served as a model for the earlier investigators in this field. Confining ourselves for the moment to equations of the first order, it may be possible to reduce such an equation to the form dx + dy 1-- -.-- 0, R(x) S(y) where R(x) is a function of x, and S(y) a function of y alone. The variables are then said to be separated, and we may write 1 dy ___ c R(x) m ' where c is an arbitrary constant. Owing to the fact, which has just been mentioned, that the problem of areas is solved by the com putation of an integral of the form) f(x)dx, a such an integration is lcnown as a quadrature. If the variables can be separated, the differ ential equation may, therefore, be integrated by quadratures.
The earlier analysts believed that any differ ential equation could be integrated by the elementary functions then in use, and by quad ratures. This we now know not to be the case, just as we lcnow, since the days of Abel, that all algebraic equations cannot be solved by the mere extraction of roots. (See ALGEBRA ;