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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

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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

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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;

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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

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and (53) has been found by dividing (52) by x2+1 { ƒ (z) +z$(z)}, which is manifestly equal to Pa+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 = 0. Let us apply this process of integrating homogeneous equations. Ex. 1. xdx+ydym (xdy-y dx).

(x+my) dx + (y-mx) dy = 0;

therefore the integrating factor is

{x2+mxy + y2-mxy}-1 = (x2 + y2)−1;
—mxy} ́

and the equation becomes an exact differential of the form

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Ex. 2. Again, let us take Ex. 2, in Art. 377, x2y dx-(x3 + y3) dy = 0;

the integrating factor is -y-4: therefore

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-1

380.] And that the factor (P+Qy) 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;

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dr

d y d dy dy (Px+Qy)2

dr

dq

dx

Q { x ( 1 ) + v ( 1 }) } − P { x ( 1 ) + y (dg) };

dy

(Px+Qy)2

dy

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since by Euler's Theorem, P and Q being homogeneous functions

of n dimensions,

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and thus the criterion of integrability is satisfied.

If P and Q are such that Pa+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

(α1x+b1y+c1) dx + (a2x+b2y+c2) dy = 0:

(57)

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(59)

and substituting in (57), and reducing,

(b2 + α, n) d§— (b1 & + a, n) dn = 0;

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,

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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

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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;

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(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

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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

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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.

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