which is linear with respect to v. The integral is SECT. 3. Equations integrable by separating the I. Homogeneous equations of the first order and degree can always be integrated by means of the separation of the variables. If the two variables be x and y, assume and by means of one of these equations and its differential eliminate one of the variables and its differential from the given equation. The resulting equation involving and the other variable always admits of the variables being separated. This method of integrating homogeneous differential equations of the first order was first given by John Bernoulli. See the Comm. Epis. of Leibnitz and Bernoulli, Vol. 1. p. 7. Ex. (1) Let the equation be If m>2, the denominator of the part under the sign of integration is of the form (≈ − a) (≈ --), and therefore Let m<2, so that we may assume m = 2 cos a. Then = C. Then the inte whence == X (1 + x2)! ' x = C {z + (1 + x2) 1}, from which x2 = 2 Cy + C2. (3) Let (y + y3) dx = 3xy'dy. Assuming y= æ≈, we find an equation which is easily integrable, since the second side is a rational fraction. The final integral may be put under the form (No2 - 2y3)3 = Cx2. (4) Let y3dx + (xy + x2) dy = 0. Assume yx, when the transformed equation becomes x = dy dz + = 0, y 28+82 (5) Let xd + ydy = m (dy – yda). Assuming y=xx, we find the integral to be (7) Let ydy + 3yxdx + 2x3 dx = 0. The integral of this is y2+2 = C (x2 + y2)3, (8) Let a2y dx - y3dy = x23dy. The integral is y-C (9) 23 (10) Let the equation be x23dy - x2ydx + y3dx - xy' dy = 0. In this case the transformed equation is reduced to x (1 − x2) dx = 0; which may be satisfied by x = 0, 1-2 = 0, or dx = 0. This last is the only differential equation, and therefore is the solution of the equation. It gives as the integral The other two solutions correspond to particular values of the arbitrary constant. The first or a=0 gives c = ∞, the second or 1 gives C = ± 1. II. Equations in which the variables can be separated by particular assumptions. (11) Let (mx + ny + p) dx + (ax+by+c) dy = 0. by means of which the proposed equation becomes (mx - nu) dx + (bu - az) du = 0, which is a homogeneous equation integrable by the usual assumption. If m n = b a this method fails, but the given equation is then easily integrable: for eliminating m it becomes b (cdy + pdx) + (ax + by) (bdy + ndx) = 0; = and by assuming ax + by whence bdy dx - adx, the equation becomes {ac-bp + (an) ≈ } dx = (c + z) dz, in which the variables are separated. Euler, Calc. Integ. Vol. 1. p. 261. (12) Let dy = (a + bx + cy) dx. By assuming bx + cy = ≈ we find the integral to be b+c (a + bx + cy) = Ce*. Euler, Ib. p. 262. (13) Let dy + b2 y2 dx = a2 x dx. Assume y", by which the equation becomes In order that this may be homogeneous we must have a homogeneous equation in which the variables are separable. This equation was first considered by Riccati in the Acta Eruditorum, Sup. VIII. p. 66, and it usually bears his name. It may be converted into a linear equation by assuming (14) If in the equation of Riccati m = 0, the variables are immediately separable. It becomes then The assumption y is not the only one which renders the equation of Riccati integrable. If we assume the equation becomes y = Ax2 + x1z, x2dz + (qx2¬1+2 b ́A x2 + 9 + b2x21) ≈ dx + (pAx2¬1 + b2 A2x2o) dx This will be reduced to an equation of three terms, if |