Secondly, assuming y., representative of any of the remaining variables y2, Y,,...y, as subject, we have the equation It follows from the above that, if we integrate the auxiliary system dx_dx, dr n = Χ. X2 .(8), the values of y2, Y.,... Yn will be the first members of the integrals of that system expressed in the form an Y2 =α, y3= a,... Y2 = α, And it follows from (6) that if, from the system ·(9). .(10), differing from (8) only in that it contains one additional member dy,, we deduce an additional integral equation connecting Y with the original variables x,,,,...,, that equation will give the value of y. We see that the number of distinct auxiliary equations is precisely equal to the number of quantities to be determined, so that the scheme is a consistent one. The solution of the problem is therefore virtually dependent on the partial differential equation (6), from the auxiliary system of which, (10), it suffices to deduce n integrals, one expressing y, in terms of x, x,,... x, the others determining Y2, Y3,... Yn, as functions of x,,,,..., in the forms (9). To the arbitrary constant in the value of y, we may give any value we please. Introducing the new variables, the equation given now assumes the form which must be integrated as if u and y, were the only variables, an arbitrary function of y,, Ys,... y being introduced in the place of an arbitrary constant. Finally, we must restore to yy,...y their values in terms of x, x2, ... „. n Here, the form of the first three terms shews that we must d +(1−xy) and the equation assumes dy (π2 + n2) u = 0. To avoid the difficulty arising from the imaginary factors of 2+n2, let us assume two new variables, a' and y', such therefore (~2 + x2) u = 0; u = cos nx'p (y') + sin nx'y (y'), or, restoring to x and y their values, u = cos {n log √(1±2)} -X y -x √(1 − x*) 10. Solve, by the method of Art. 10, the equation may be reduced to that of two linear equations of the first order. CHAPTER XVII. SYMBOLICAL METHODS, CONTINUED. 1. THE classes of equations considered in the last chapter might all be gathered up into the one larger class represented by ƒ (π) u = X, being a symbol combining with constant quantities as if it were itself a symbol of quantity. But linear differential equations do not, except under particular conditions, admit of expression in this form. Those which are of the ordinary species involve in their general expression two symbols, x and d operating in combination on the sought and dependent dx' variable y; and no substituted form of such equations is general which introduces fewer than two symbols in the place of a and. We propose in this chapter to employ a trans d dx formation which is general, and which is adapted in a very remarkable degree to the development of general methods of solution. A somewhat fuller account of it will be found in a memoir on a General Method in Analysis (Philosophical Transactions for 1844, Part II.). Other principles and other methods will also be noticed. The following theorems, demonstrated in Chap. XVI., will frequently recur. |