Since the surface represented by the general integral is the envelope of a single infinity of surfaces represented by the complete integral, each of these latter surfaces will touch the former along a certain curve; such a curve is known as a characteristic. If the partial differ ential equation is not linear in and a ay there are characteristics. A linear equation has only characteristics. The integral sur faces may be looked upon as generated by characteristics, and the usual method of inte grating the partial differential equation con sists in setting up a system of ordinary differen tial equations which determines the character istics.
The points of view in the higher theory. In speaking of ordinary differential equations, we have already mentioned the fact that the point of view of the elementary theory is inade quate even in those cases in which the reduction to quadratures is possible. Given for example, the equation ( 1 34) ( I 18 which may be reduced to a quadrature, dy x=fv(i_,(,-k23+2).
The reduction of the equation to this form is a mere formal process which, in itself, teaches us nothing. We shall have to ask ourselves the following questions: to what extent does a given differential equation define a function y of x? what are the characteristic properties of this function? what analytical processes in volving known functions, infinite series, prod ucts, etc., will serve for the computation of the values of the function for all of the values of its argument? In the case of the above differ ential equations these questions have been com pletely answered by the creation of the theory of elliptic functions by Abel and Jacobi. In gen eral it is to be expected that every differential equation defines a transcendental function; it is the theory of these transcendentals which constitutes properly the most important por tion of the theory of differential equations.
In order fully to understand the properties of functions it has been round necessary to look upon the variable as being capable of assuming not only all real but also all complex values. In the hands of Cauchy, Riemann, Weierstrass there has grown up in this way the theory of functions of a complex variable (q.v.). This theory serves as a base for our further discus sions. We shall, however, confine ourselves to a few of the simplest cases, merely indicating the general point of view.
Let dy be the given differential equation: Let f(x, y) be analytic in the vicinity of (xo, yo) i.e., let it be possible to develop f(x, y) into a series proceeding according to positive integral powers of x xo and y yo. Then, as was first proved
by Cauchy, there exists a function y of x which may be developed according to positive integral powers of s which reduces to y = ye for so, and which satisfies the differential equa tion. This theorem, which may be easily gen eralized to apply to equations of higher order, or to systems of equations of the first order, is generally known as the fundamental theorem of the theory of differential equations. It proves the existence of analytic functions which are uniquely defined as solutions of analytic differential equations and which satisfy the sub condition of reducing to given values for a given value of the argument. The theorem may be proved by the method of dominating functions. This consists in finding a series which formally satisfies the differential equation and reduces to yo for x= so; its convergence is then demonstrated by comparing it term for term with a corresponding series, which is formed in the same way from another differential equa tion, and which is known to be convergent. The exact circle of convergence cannot, however, be generally stated. A great many papers have been written on questions which easily suggest themselves in connection with this theorem. If the function f(x, y) is not developable in the given form; if, for example, its development contains negative or fractional exponents, how far are its solutions determined and what is the form of their developments? Besides the ana lytic solutions whose existence Cauchy has demonstrated, are there other non-analytic solutions? The first investigations of these questions are due to Briot and Boquet. They have since been completed by a great many authors.
Cauchy's existence theorem can be made more precise in the case of linear differential equations. Let dny dnly (27) +p.-+
dxn
be a homogeneous, linear differential equation of the nth order. In the vicinity of x = so let the coefficients pa be expressible as power-series, proceeding according to positive integral powers of x so, and convergent for all values of x for which Is