Existence Theorems and General Theory - equations, series, interval, paris, solution, conditions and equation

EXISTENCE THEOREMS AND GENERAL THEORY On the theoretical side the first question that presents itself in connection with a given differential equation concerns the existence of solutions. Suppose that we have the equation dy — =f(x, y), dx where f(x , y) is an analytic function of x and y that is regular in the neighbourhood of the point Then the existence of an analytic solution y = y(x), that is regular in the neighbourhood of the point x = and takes the value when x = may be established as follows: From (24) we get by successive differentiations of of dy ax + ay dx In the right members of (24) and these successive equations we put for x and for y, and use the resulting values of the successive derivatives of y to form the series dy ( + d?2l (x + • • \dx / 2.

0 0 0 n! (x — xo) n dxn/ • • • , (25) If this power series converges within a certain interval for x, we know from the theory of power series that it can be differentiated term-by-term within this interval. Hence But the series in the right member of this equation is the expan sion of the analytic function in the right member of (24) after y has been replaced by the analytic function 4(x). Moreover, We have therefore in (25) a solution of (24) that satis fies the given conditions, provided that the series in the right member of (24) converges within some interval. The establish ment of this convergence is the crux of this proof of the existence of an integral. It is accomplished by recourse to what Cauchy (q.v.), to whom we are indebted for this proof, called the "Calculus of Limits." The essential feature of this method consists in setting up, a so-called majorant function of whose convergence we are assured. This majorant function is represented by a power series each of whose coefficients is positive and not less than the absolute value of the corresponding coefficient in (25). We cannot give the details here. The solution obtained in this way is obviously unique.

Picard established the existence of these solutions by making use of a method of successive approximations. In his proof it is not necessary that f(x, y) be analytic. It is sufficient to as sume that for Ix —xo l

The successive approximations are obtained from the fol lowing equations dx =f(x, yi-1) (i> I), subject to the condition that = By integration we get ff(x, yo)dx+yo, x y= = f (i> I) Now the proof consists in showing that for a sufficiently small all the functions satisfy the inequality I —yo I

Each of these existence proofs suggests a method for obtaining an approximate solution of the equation.

Only slight modifications need to be made in these proofs to have them applicable to systems of n equations of the first order in one independent and n dependent variables. And since an ordinary equation of order 11 is equivalent to a system of n equa tions of the first order of the kind described, it follows that these theorems also apply to ordinary equations of order n. For such an equation there is, therefore, a unique solution y such that y and its first i derivatives have assigned values for x = provided that the conditions set forth in the existence theorem are satisfied.

In the preceding discussion we have confined our attention to the domain of real variables. There is, however, at times an ad vantage in including the domain of complex numbers (q.v.). In this more extended domain the proofs of the existence theorems given above are valid, provided that in the second one we re strict the functions to being analytic.

We add a brief sketch of two points in the general theory. These concern equations of the second order of the form y"+ [o(x)+Xp(x)]y=o, where o•(x) and p(x) are continuous and p(x) > o in the interval We wish to know whether there are solutions y of (28) that satisfy the two conditions Y(xo) = o and y(xi) = o. Heretofore we have imposed on the solution the condition that it and its first derivative shall have definite values at the initial point, whereas here the conditions prescribe the values of the solution at the two ends of an interval. It is for this reason that the problem is described as a "boundary-value problem." There are, of course, more general boundary-value problems.

Now it turns out that (28) has a unique solution that satisfies conditions (29) if, and only if, X is one of an infinite set of values x2, • • • , x , • • • . These are called the characteristic values, and the unique solutions • • • , • • • to these are called the characteristic functions, of the problem.

If in (28) p(x) = I, o, and 7r, it can readily be seen that = and = sin nx, where n is a positive integer. Now it is well known that functions satisfying certain conditions can be represented in the interval (o, 7r) by the series This is merely a special case of Fourier's series (q.v.), which, in its turn, is a special case of an important and general theorem concerning the characteristic functions of equation (28) with the boundary conditions (29). This theorem is as follows: Any function f(x) that is continuous in the interval together with its first two derivatives, and vanishes at the ends of the interval can be represented in this interval by the uniformly convergent series plied by a properly chosen constant factor.

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