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. If m>2, the denominator of the part under the sign of integration is of the form (≈ − a) ( a and therefore Let m<2, so that we may assume m = 2 cos a. Then = C. y cos a mx + x2 = (1-x). Let m = 2, or 1 gral of the equation ⚫becomes log (xy) C or = (2) Let xdy - ydx = (x2 + y2)§ dx. Making y=xx, this becomes whence x = C {≈ + (1 + x2) } }, from which a2 = 2 Cy + C2. (3) Let (x2y + y3) dx = 3xy❜dy. Assuming y=xx, we find Then the inte an equation which is easily integrable, since the second side is a rational fraction. The final integral may be put under the Assuming y=xx, we find the integral to be Assuming yx, we find as the integral y = = (7) Let y3dy + 3y3 xdx + 2x3 dx = 0. The integral of this is y2 + 2x2 = C (x2 + y2)§. (8) Let a2y dx - y3 dy = x3dy. The integral is y-c (9) Let Let xydy - y° dx = (x + y)" e ̄* dx. - € The transformed equation is 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 = 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) d x + (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 = ds - adx, the equation becomes {ac-bp + (an) ≈ } dx = (c + ≈) dx, 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) = C€oo. Euler, Ib. p. 262. Assume y=x", 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 + x1≈, x1dz + (qx1−1+2 b ́Ax2+9+b2x2o) ≈ d x + ( p A x2¬1+b2Ã2x2o) dx = a2x dx. This will be reduced to an equation of three terms, if we have |