dx - dy {ag+ax2 +24x4 + 2928]} {ac+azy2 + 2494 +2698}t' by assuming x2 = , y2 = n; and more generally dx dy x{ag + Qqx" +0g 22n} y{ao+azy" +azy2"}}' by assuming x* = $, y" = n. These questions however are only particular cases of the general theory of the new elliptic and other functions; which is a subject requiring distinct discussion, but lying beyond that proposed in the present work. 422.] Some functional equations also are conveniently solved by means of integration and differentiation, as the following examples shew. Ex. 1. Determine the form of 2 = f(x), so that for all values of x and y f(x) + f (y) = f (x+y). Taking the x-differential, f'(x) = f'(x+y); whence, y being independent of x, we infer that f'(x) is constant whatever value x has; therefore f'(x) = c, f(x) = cx+1; substituting which in the given equation, CX+61 + cy +C1 = c(x+y)+C; .. c = 0; and therefore the most general form of f (x) which satisfies the equation is f(x) = cx. Ex. 2. Find the form of f, so that f (x)+f (y) = f (xy). Taking the x-differential, f'(X) = yf'(xy); again taking the y-differential f'(y) = x f'(xy); i. æf'(X) = yf'(y); consequently xf'(x) is a constant: let us suppose x f'(x) = a; .. f(x) = substituting which in the given equation, a logo + a log? = a log mily; :: c=1; .. f(x) = a log x. Ex. 3. If f(x)f (y) = f(x+y), f(x) = ear. PRICE, VOL. II. 423.] The differential equation Puý'+P2Y+Pg = 0, where P1, P2, P, are functions of x, has been integrated in Art. 382. The form which next suggests itself is P, ý' +PzY+Pgy+PA = 0, where P, ... Pe are functions of x: but this has never yet been completely integrated, and will not be, until the properties of certain transcendents, which are in the form of definite integrals, have been more completely investigated : a particular form how. ever of it is + ay2 = bx”, (251) which is known by the name of Riccati's Equation, having been discussed by Riccati in the year 1775 in the Acta Eruditorum, and of which, in particular cases, solutions can be found : these I proceed to investigate. First suppose m = 0; then (251) becomes dy (254) and (251) becomes x?dz +aza dx = bxm+4 dx ; in which equation the variables are separated if m = -4; and we have do -0. az? - 7 + m2 = 0. Again, in (254) let v = 5; then dz-azo du = -bu-m-4 du, which is of the same form as (251); and therefore if (251) is integrable for any particular value of m, say u = m, it is also integrable when m = -u-4. 424.] Again, in (251) let y = ; then dz = a dx—bzoxm dx. (255) Let (m + 1)2" dx = dv; then (255) becomes dz 6 vm+1; m +1 which is of the same form as (251); and therefore if (251) is integrable for any particular value, say y, of m, it will be integrable also when (256) ut1. Now we have seen above that (251) is integrable, when u = -4, therefore the equation is also integrable, when a je Also from the conclusion of Art. 423 we infer that the equation is integrable when and therefore from (256) it appears that m = - m = —~—4, m= 4; we have the following series of values : 12 - .... (257) (257) 73..... 4 8 12 16 - un the types of the general terms being respectively - 2- and -4n ; and in which if n = 0, and if n = 0, we have the two values of m, viz. 0, and -2, which on inspection render (251) integrable. y (x+3x8) +3 y(xt – 3x})+3 425.] This example, and it is one of the easiest, sufficiently indicates the tediousness of the process, and the succession of the substitutions. If m has a value corresponding to the first term of the series (257) the method is of course that of Article 423: but if m has any other value, then we shall have to pass successively by alternate processes from one series to the other, until at last we arrive at a form wherein m has the value - 4. The above process is unsatisfactory, because although it points out certain cases where the variables are separable, still the number of them is limited ; and they are obtained by particular artifices, and the investigation does not prove that they are the only possible ones. M. Liouville, however, in the VIth volume of his Mathematical Journal, has proved by a rigorous investigation that the cases comprised in the above series are the only ones where the integral can be expressed in an algebraical, logarithmic, or exponential form. There are also other forms which are capable of reduction to Riccati’s Equation. Thus, if dy + aya x" dx = bx dx ; let x"dx = dz; : x*+1 = (n+1)z; dy + aya dz = b (n + 1)n+1 zn+1 dz, which is of the form (251). The Equation of Riccati also admits of transformation into a differential equation of the second order, under which it is often convenient to consider it. + ayo = bx". Let - 1 dz: : dy - 1 dz? l daz. dir we have dita + 7 dt = lz ; and if z= ut-", do u n (n-1).. t = lu. dt? All these therefore are equivalents of Riccati's Equation; and the properties which are true of any one are also true of each of the others. If therefore we can determine either a particular or a general integral of either, that of Riccati's equation will be determined by the equation log z = / y dx. A Memoir by M. Malmsten of the University of Upsala, and inserted in Vol. XXXIX. of Crelle's Journal, p. 108, on the various forms and properties of Riccati's Equation, may be consulted with advantage by the reader who is desirous of further information. |