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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
dyn =P (%); and since then = tant, and = tan 0, 7 and 0 being the angles 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 if r = 20; then
dy 2 xy
dix = x2–42;
.. logo + log (22+1) = 0;
x2 = 2cy - y2; 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 ms 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 som = tan ot'M;
. . y - x dx .
.. wydy +(x2 - y2) dx = 0;
i mod = czeka. 379.] By the introduction of the new variable 2 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
380.] And that the factor (Px + 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
and thus the criterion of integrability is satisfied.
If P and Q are such that Px+2y = 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
(a,x+by+C) dx + (aq x + b2y+ca) dy = 0: (57) let 2,x+by+c = $, 2, x + b2 y +c, = n;
:. df = a, dx +b, dy, dn = a, dx + b, dy;
az dę – a, dn .
a, b, - a, ba'
(59) a homogeneous equation, which is integrable as above.
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, unless
(60) ā, = b. = k (say),
SECTION 4.—The integration of the first linear differential
equation. 382.] Another form of differential equations in which the variables admit of separation is
Pudr + P2Y+Pz = 0, where P1, P2, P, are functions of x only; and which is called the linear equation of the first order, because my and y enter in only the first degree, and there is a vague analogy between it and the equation to a straight line.
Dividing through by P, and making obvious substitutions, the equation becomes dy + f(x) y dx = F(x) dx.
(64) Let y = zt; a. dy = zdt +tdz ;
... z dt+t dz+f(x) zt dx = F(x) dx
(66) 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,
dz+f (x) > dx = 0; .
(68) thus (66) takes the form dt = e$1(x)dx F(x) dx ;