II. If the equation be of the first order and homogeneous in and y, and if we assume y=ux or x=uy we shall obtain by the elimination of the variables an equation dy between u and da which combined with the differential of ux or uy will give us the means of finding the relation between x and y. This equation is homogeneous in x and y, and may be treated like the preceding examples, but it is more convenient to proceed as follows. Square both sides, and solve the Whence, dividing by the second side of the equation and III. Equations integrable by Differentiation. If y = xp +ƒ (p) we have, on differentiating, where p = dy), 0 = {x +ƒ'(p)} dp. This is satisfied by dp = 0, or y = Cx + C', where C' = f(C). The singular solution is found by eliminating p between the given equation and x +ƒ ̃(p) = 0. This equation is known by the name of Clairaut's form, having been first integrated by him. See Mémoires de l'Académie des Sciences, 1734, p. 196. The general integral is y = Cx + n (1 + C2)3 ; the singular solution is (11) Let x2 + y2 = n2. y = px + p-p2. The general integral is y = C (x + 1 − C); the singular solution is 4y = (1 + x)2. (12) Let y - px = a(1 - p3)3. The general integral is y = Ca + a (1 − C13)1 ; a The general integral is y = C{x + c)}; the singular solution is 3+ y3 = a3. Sometimes an equation which is not of Clairaut's form may be reduced to it by being multiplied by a factor. Multiply by 4y, and let y2 = u, and 2y dx = du. Then a + (4 x − 2b) · 4u = 0, On multiplying by 4xy, and taking a and y as the new variables, the equation becomes of Clairaut's form, and the integral is The singular solution is a + b (x* − a3)2 = 0. If (x2 where P and Q are both functions of p, we have by differentiation which being a linear equation in P, may be integrated, so that and as y = [pda, we can we have a expressed in terms of Substituting in this the value of p derived from the equation, we have the required integral. By eliminating p between this and the given equation, the integral is determined. dy da r and y to be both of one dimension, will be of o dimen sions, and will be of 1 dimensions. The equation d2y da then is said to be homogeneous when, adopting this scale, the sum of the indices in each term is the same. integrate an equation of this form, let dy dx To d2y = P, = d.x the quantity a can be eliminated so as to give a relation between u, v, and p. But as dy = pdx = udx + xdu, we have = and as dpqdx, we have also vda adp. From this may be eliminated by means of the given equation, and we have a differential equation of the first by integrating this we obtain p |