The integral of this is Monge's Method equation m? – Pm + R = 0; and if from these two systems we can find two integrals U = a, V = b, then V = f(U) is the first integral of the proposed equation ; and the integral of this is the complete integral of the proposed equation. It is generally more convenient (when possible) to find another first integral, of the from . . . v'= f,(U'), and between these to eliminate p or q so as to obtain an equation involving only one differential coefficient, and which is therefore easily integrable. Monge, Mémoires de l'Académie des Sciences, 1784, p. 118. the integral of which is ..(24- a) (- 9) + 2x = b = f (y - x), and therefore x + y + y From the first of (2) we find y + x = ajo and substituting this in the equation just found, it becomes dx dx 2x _ f (y – x) dy dva a This is a linear equation, and is therefore easily integrated. The result is aj where x + y is to be substituted for a, after integration. (12) Let the equation be If we put p + q = a, this takes the form represents the integral of the proposed equation. Poisson* has shewn how to obtain a particular integral of equations of the form P = (rt - 52)" Q .................. (1) where P is a function of p, q, r, s, t, homogeneous with respect to the last three quantities, and Q is a function of x, y, x, and the differentials of z, which does not become infinite when rt - s = 0. * Correspondance sur l'Ecole Polytechnique, Vol. 11. p. 410. If we assume q= f(), we have 8 = rf'(p), t = 8f' () =r{f'(p)}? ; and therefore rt – gể = 0. ............ (2) Hence the equation (1) is reduced to P= 0; and on substituting in it the values of 9, 8, and t, the quantity go will divide out, as P is homogeneous in r, s, and t, and the equation is reduced to the form F{p, f(p)f' (p)} = 0, which is an ordinary differential equation, and being integrated determines the form of f (p) involving an arbitrary constant. The partial differential equation q = f (p) can always be integrated, and furnishes a value of x involving an arbitrary function and an arbitrary constant. This process comes to the same as finding what developable surfaces satisfy the equation (1). (13) Let go? - t = rt - $?. gel {1 - [8"(p)]'} = 0, whence ľ (p) = = 1; and therefore q= f (p) = + P + C, C being an arbitrary constant. On integrating this we find %= Cx + (y + x) as a particular integral of the given equation. (14) Let t + 2p8 + (p? - a) r = 0. .............. (1) In this case Q = 0, and on putting 9 = f () we have, after dividing by r, {f'(p)}" + 2 pf'(p) + p – a' = 0,; .........(2) from which f' (p) + p = = a, and therefore q+p ap = C........ ..... |