The integral of this equation is deduced from that in d Ex. (13) of Chap. V., by putting a This gives dy 2 2 y y' f (y – ax). for q. (23) The equation d"% pm dix dxn-1 dye 0, dum dy may be integrated by the same method as that in Ex. (14) of d? Chap. vi., by changing k into – a and putting arbitrary functions of y instead of the arbitrary constants. Thus if m = 2, we have X = F (y + ax) + f (y - ax), so that the integral of 2p dx P 1 % = w2p+1 = 0 da* dy {F (y + ax) + f (y – ax)}. dx 4 de a? dv adx dy is x=3{F(y+aw)+f(y-ax)}- 3av {F'(y+aw), f'(y-ax)} + a’x{F"(y + ax) +f" (y – ax)}. (dx do % = 0. dx dy dy) (x + y)2 Assume y + x = U, y - x = 0, when the equation becomes 2 dx 2 du? dv2 u The integral of this by Ex. (18) is {$'(u + v) +%'(u – v)} - {p (u + v) +y (u – v)}. Hence, 1 dv 1 {p' (y) + V (2x)} {Q (24) ++ (2x)}. (x + y) dy 0 ().R, dy P, Q, R being functions of x and y, may be transformed into linear equations by assuming d: = d.x'. ху + y dc dz dy We might with advantage have applied the same transformation to the equations in examples (1), (3), and (4), as it is generally convenient to reduce the factor of x to two terms. Sect. 3. Equations involving the differential coefficients of . in powers and products. dx dx dy dp dy d x If the equation be of the first order make P, = 9, dx and from the given equation find q in terms of p, x, y, z, and substitute this value in the equation dp dq d q +9 р 0, (1) d & d x which will then become an equation of the first order between four variables. The value of p found by integrating this, with the corresponding value of q will render dx = pdx + qdy, (2) a complete differential, and this being integrated will give the value of s. The integral of the first equation will involve an arbitrary constant (a); and the integral of the second will introduce another (6), which is to be considered as an arbitrary function of (a); and we shall thus obtain an integral of the form f (x, y, x, a) = q (a), from which a is to be eliminated when a specific meaning is assigned to 0. Lagrange, Mémoires de Berlin, 1772, p. 353. (1) Let p + q* = 1, or q = (1 - po), р dp da P dp (1 – po)! dz Substituting these values in equation (1) it becomes dp dp dp + (1 - p) dx This equation is integrable if we can integrate the system of equations d p=0, pdx - dx = 0, (1 – po)?dz - dy = 0. da dx + P da = 0. dy so that The first gives p=l, whence q = (1 – a”), and dă = adx + (1 - a')!dy, z = ax + (1 - a') y + P(a). If we differentiate this with respect to a we obtain the equation (1 ( = X - y + $'(a), between which and the preceding we may eliminate a when o is specified. (9) Let Pa = 1. 2 dp p2 dx Y + + 0; pdz a = ax + aʼy + (a). (6) Let q = x P + p. The integral is x = x 6%++ 62 (4+4) + (a). 'do ed & = c". d x = c". By assuming d' , ' +; the equation becomes 'do dy The integral of this found by the same method as in Ex. (2) is mtnty m n 1 + m-a m +n n-B a" xm y m+n+y + + (a). n-ß an when m +n + y = 0, z' = log x. |