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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;
Let x = yz; i. dx = ydz+z dy;


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.. 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 .

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.. 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

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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

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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;
a b2 dę – b, dn

az dę – a, dn .
, dy = unur; (58)
a, b,-a, b,

a, b, - a, ba'
and substituting in (57), and reducing,
(62 $ +22n) d&76,8+a, n) dn = 0;

(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),

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SECTION 4.The integration of the first linear differential

equation. 382.] Another form of differential equations in which the variables admit of separation is

I dy

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
zdt+t{dz + f (x) z 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; .
· log z=-[f(x) dx,
z = e-$f(x) dx;

(68) thus (66) takes the form dt = e$1(x)dx F(x) dx ;



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