Although either of the substitutions y = xz, or x = yz, will produce the result, yet a judicious choice will frequently shorten the process the student however must in this matter be left to his own skill. 378.] Let us also consider homogeneous equations of the form (52), and the introduction of the new variable, from a geometrical point of view; (52) may evidently be put in the form respectively at which the tangent to a curve and the radius vector are inclined to the axis of x, the above equation, interpreted geometrically, expresses a relation between these two angles. Thus ifr 20; then which is the equation to a circle, whose radius is the arbitrary constant c. And to take another example, see fig. 50: to find the equation to a curve such that a perpendicular мs let fall from M, the foot of the ordinate, on the radius vector OP shall cut the axis of y at the point T', where it is cut by the tangent PT. In this problem tan soм y = tan OTM; x dx x y dx - xdy .. xy dy+(x2 — y2) dx = 0; 379.] By the introduction of the new variable z the original expression (51) has been so transformed that the variables admit of separation; let us examine the process more closely take the form (52) and compare it with (51); then and (53) has been found by dividing (52) by "+1 {ƒ(z) +z¢ (z)}, which is manifestly equal to P+Qy; hence the equation (51) satisfies the criteria of integrability when it is divided by Px+Qy; therefore (P+Qy)1 is an integrating factor of p dx+Q dy Let us apply this process of integrating homogeneous equations. Ex. 1. adr ydym (xdy-y dx). -1 (x+my) dx + (y-mx)dy = 0; therefore the integrating factor is {x2+mxy + y2 —mxy}−1 = (x2 + y2)−1 ; and the equation becomes an exact differential of the form Ex. 2. Again, let us take Ex. 2, in Art. 377, x2y dx-(x3 + y3) dy = 0; the integrating factor is -y-4: therefore x2y dx-(x2+y3) dy = DU; = 0. 380.] And that the factor (P+Qy)1 renders (51) an exact differential is also evident, inasmuch as the condition of integrability becomes satisfied: for multiplying by this factor, we have Pdx+Qdy = 0; since by Euler's Theorem, P and Q being homogeneous functions and thus the criterion of integrability is satisfied. If P and Q are such that P+Qy= 0, the preceding process fails; in this case however another integrating factor can be found by a method which will be developed in a future section of the present Chapter. 381.] A form of differential equation which is easily reduced to the homogeneous form is (a1x+b1y+c1) dx + (ax+by+c2) dy = 0: (57) and substituting in (57), and reducing, (b2 ¿+ a2 ŋ) d§—b1 § + a1 n) dŋ = 0; a homogeneous equation, which is integrable as above. (59) This transformation is manifestly equivalent to that of a system of rectangular coordinate axes, in which the origin and the direction of the axes are changed and it is always possible, : for in this case dέ and dŋ are infinite: but then (57) becomes (ka2x+kb2y+c1) dx + (α2x+b2y+c2) dy = 0, (a2 x+b2 y) (dy+ k dx) + c2 dx + c2 dy = 0; 2 (61) in which, if we put a x + b2 y=2, and eliminate x or y, the variables will be separated, and the integration can be performed. Also by a similar substitution may the variables be separated in the equation dy = f (ax + by) dx. (62) SECTION 4.-The integration of the first linear differential equation. 382.] Another form of differential equations in which the variables admit of separation is where P1, P2, P3 are functions of x only; and which is called the dy linear equation of the first order, because and y enter in only dx the first degree, and there is a vague analogy between it and the equation to a straight line. Dividing through by P1 and making obvious substitutions, the equation becomes dy+f(x)y dx = F(x) dx. .. dy = ≈dt+tdz ; (64) Let y = zt; (65) ..zdt+t dz+f(x) zt dx = F(x) dx z dt + t {dz+f(x) z dx} = F(x) dx. As we have introduced two new variables z and t, and have made only one supposition respecting them, we may make another; let this be, (66) dz+f(x) z dx = 0; (67) ... log z = - ff (x) dx, z = e-ff(x)dx; thus (66) takes the form dt = eff(x) dx F(x) dx; (68) (69) No constant has been introduced in (68), because it is desirable to keep complex formulæ in as simple a form as possible: and the generality of the final result is not affected by the omission, for such a constant would disappear in (69) by reason of the form of the result. In terms of definite integrals (69) is y = e−f;%f(x)dx { % + [*es";f(x)dx y (x) dx} . Ex. 1. dy+y dx = e-* dx. F Ex. 3. (x+1) dy-ny dx = e* (x+1)*+1 dx. y = (e + c) (x+1)". Ex. 4. 2dy + 2y cos x dx = sin 2x. 383.] A form which admits of reduction to (64), and consequently of having its integral determined in the form (69), is dy dx which is generally known by the name of Bernoulli's linear equation of the first order; for dividing through by y", we have Let y ̄"dy+y−n+1f (x) dx = F (x) dx. y-n+1 = 2; ..(n-1) y "dy = dz; and therefore by substitution, and by (69), (71) dz-(n-1)zf(x) dx = −(n−1) F(x) dx ; (72) 1 2= -(8-1) = e ̄(n−1) ƒƒ (x) dx F {c—(n−1) fe("−1) ƒƒ (x) dx y (x) dx}. (73) The explanation of the failure of the above substitution when n=1 is too obvious to require more than a passing remark. |