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and the requisite condition that the second member should be

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Ex. 2. (xy-ex3) dx-x2y dy = 0.

394.] Again operating on the first two members of (134) in the manner indicated by the following results, we have

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If Px+Qy = 0, then from (139) we have also

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which is a solution of the differential equation; or in an apparently more general form,

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Pdx+Q dy

=

0, and P+Qy = 0.

If qy-Px = 0, then from (140) we have also

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(142)

(143)

which is the solution of the differential equation; and in an apparently more general form, we have

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Equation (143) is also true because we have simultaneously QY-Px = O, and Pdx+Qdy = 0.

395.] If however neither P+Qy nor Qy-Pa vanishes, then from (139), we have

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now the right-hand member of this equation is an exact differential if P and Q are homogeneous functions of x and y of n dimensions. For in that case by Euler's Theorem, we have

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=

2 {(d) dz + (de) dy} + y { (d) dx + (de) dy} = 0;

dx

dy

and adding this to the numerator of the right-hand member of (145), and observing that

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hereby then can u be determined, and an integral be found. Let these results be compared with Articles 379 and 380. If both members of (147) are multiplied by F'(u), then we have

r(u) = f¥'(u) du = f3dx+Qy

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and therefore if one integral of (147) can be found, an infinite

number may also be determined.

Hence, if P and Q are homogeneous functions of n dimensions, (Px+Qy) is an integrating factor of Pdx+qdy = 0. The following are examples of integration by this process.

Ex. 1. 2xy dx + (y2x2) dy = 0.

Here P+Qy = x2y + y2;

2 xy dx + (y2 — x2) dy

= DU

0;

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x2 + y2

=

log

;

y

= c,

y

where c is an arbitrary constant.

Therefore by reason of (148) the most general integral is

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the right-hand member of which is an exact differential if P = yf(xy), Q = xp (xy); for in this case

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And thus if P = yf(xy), Q = x(xy), (Px—Qy)-1 is an inte

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396.] For another illustration of the theory of integrating factors let us take the first linear differential equation which is given in (64), Art. 382, and a means of integrating which has already been therein discussed. The form is

{y f(x) − F(x)} dx + dy

= 0.

(152)

On comparing this with Art 393, it appears that the condition requisite for the existence of a factor which is a function of x only is satisfied, so that we might deduce the form of the factor from that article. Let us however investigate it from first principles.

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multiplying (152) by this value of μ we have

eƒƒ (x) dx {y ƒ (x) — F(x)} dx + eff (x) dx dy = 0;

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-

= [ eff(x)dx {yƒ (x)—r(x)} dx ;
= yeff(x)
yeff (x) dx ____
— feff(x)dx y (x)dx ;
= Sessar dy

,, =

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(153)

(154)

(155)

(156)

(157)

where c is an arbitrary constant, and is the second arbitrary constant introduced in the integration of the series of equations given in (154); so that as c, = '(c), where ' denotes an arbitrary function, from (155) we have

H = essundee { yessuar — [essinary(x)dx}{

dx

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(158)

And applying this most general value of μ, we have as the general integral of (152)

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· { yes/(x) dx — [ ef/(x)dx p (x) dx } = c. (159) —

It appears then that the equation (152) when multiplied through by eff(x)dx is an exact differential, and may be integrated as such; this is also otherwise evident; since

dy+yf(x) dx = F(x)dx ;

eff(x)dx dy+yeff (x)dx f (x) dx

= eff(x)dx F(x)dx;

(160)

whence, as the left-hand member is an exact differential, by integration we have

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which result is the same as that of Art. 382.

The following are examples of this process.
Ex. 1. dy+y dx = ax" dx.

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[f(x)dx =

= x;

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(161)

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.. y = a {x" — nx"-1 + ... (—)"n(n-1)...3.2.1}+ce*,

where c is an arbitrary constant.

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