+ 6d'. - 3d. dy: + 6d? The conditions that v should be the third differential of a function d^-3u, are du du do do - 3d. 10d3 + &c. = 0 ... (5), dx, dx dx, dx, dv dv du dv 10d3 + &c. = 0... (6). dy dy dys In a similar manner are found the conditions that v should be a differential of any order : the numerical coefficients follow the law of those of the Binomial Theorem in the case of a negative index. These remarkable formulæ were first discovered by Euler (Comm. Petrop. Vol. vir.) in his investigations concerning maxima and minima. A more direct demonstration is given by Condorcet, in his Calcul. Integral. Ex. (1) Let v = d’u = a d’y – yd’a. dv dv dv Then = d'y = y2 = 0, y dx dx, Ya - d’y = 0, the second condition is also satisfied, and we find du = xdy - ydx + C. (2) Let V = du = a* ďy + (a + 2) x dy dx + (ay + 2x) dx* + (axy + x^)d’x. Both the conditions (1) and (2) are satisfied in this case, and we find du = m dy + axyda + x? dx + C. (3) Let v = d’u = (a x – 2y) d'y – 2 dy' + 2a dy dx + ayda x. In this case the conditions (3) and (4) are both satisfied, so that v is the second differential of a function, which is found to be u = a ry - y2 + C. or dx + dy dy dx + Rd ? + Sdx dy + Tdy should admit of a first integral. If we assume S = S, + S., this may be put under the form (Rdc + Sidy) d x + (S.d x + Tdy) dy; and in order that it may admit of a first integral, we must have d d do dl dy. dy dx do But from the indeterminateness of dr, and dy this involves the conditions dR ds d T da dy d” T. d's, Whence dxdy' d x2 drdy and therefore d’R d? T d’S, ds ds + dx dy dx dy which is the required condition. The complication of the formulæ when the order of the differentials rises above the second renders their application almost impracticable, and as the subject is not one of any practical importance, it is unnecessary to adduce other examples. dS dy + 9 CHAPTER IV. INTEGRATION OF DIFFERENTIAL EQUATIONS. Sect. 1. Linear Equations with constant coefficients. THESE form the largest class of Differential Equations which are integrable by one method, and they are of great importance, as many of the equations which are met with in the application of the Calculus to physics are either in this shape or may be reduced to it. Let day d-sg dy + A + A + &c. + A-1 + Any = X, (1) dra dan be the general form of a linear differential equation with constant coefficients ; A1, A,... A, being constants, and X being any function of x. On separating the symbols of operation from those of quantity this becomes d? d d y= X, (2) dx -2 d x {a) + as we may write it for shortness. Now by the theorem given in Ex. 5 of Chap. xv. of the Differential Calculus, d the complex operation f G is equivalent to da an, Q, ... A, being the roots of the equation f(x) = 0......(3). Hence performing on both sides of (2) the inverse pro cess of sl) or C--) 6.-...---.), we have X. (4) d d d - an dir The result of this transformation is different according to the nature of the roots of (3). Ist. Let all the roots be unequal ; then by the theorem given in Ex. 6. Chap. xv. of the Differential Calculus, the equation (4) becomes d a X + ... dir -1 1 where N = (a, - a.) (a, - ay) ... (a, - (,) and similarly for the other coefficients. But by the theorem in Ex. 11 of the same Chapter, d X = 6" €-$* X = 6922 sdx e-9,8 X. dx A similar transformation being made of the other terms, - 1 dx) we find V = Neo • Jac 6-4, X+ N 49" /de 6 -12 X + &c. + N,€sdx € -4,2 X ......... (6). It is to be observed that each of the signs of integration would give rise to an arbitrary constant; and that this must be added in each of the terms when the integrations are effected. The value of y would then appear under the form y = N, 61X (Sdx e-,* X+C) + N, 6°2x (sdx e-azt X+C,) + &c. + N, 6°»X ( dx e-an* X + Cn) ...... (7). C, C2 ... Cn being the arbitrary constants. The functions Cear which arise in the integration are called complementary functions. 2nd. Let r of the roots of the equation (4) be equal to a. Then by the Theory of the decomposition of partial will give rise fractions we know that the factor di to a series of r terms in (5) of the form d ; dv the coefficient M, being equal to - a 1 when % = a. f (x) d Now (8) de </ مسی - " ( - ) a or, introducing the arbitrary constants which arise from the integration, d ed dx +* (C's + C'; x + &c. + C'1-110P -). Therefore the complete value of y is y = q** {M, s'dx" (e-4* X) + M,-1/"-dw-(6-4* X) + &c. + M Sdx (6-4x x)} + N, ed* [de (e-diox) + N, Edg* de (e-da*x) + &c. + Nn-Earrit Sdx (e-Up-** X) + €4* (C'. + C'x + C', 2+ &c. + C',-12*—") + C, eoqx + C, ezit + &c. + Cm-m. (8) There are in all exactly n arbitrary constants as there ought to be. 3rd. Let there be a pair of impossible roots, which must be of the form a + (-)*ß and a - (-)"B; then the coefficients of the corresponding terms in (6) are of the forms 1 1 and DB |