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which is the required integral. (21) Let

, 4 , du d y vi da dri - dx

9 + y = 0.
The integral is

c (C – 1) x+ + {2y + (C" – 1) x}}

(C + 1) ai – {2y + (C" – 1) x}"
There is also a singular solution y = C'x.

Sometimes an equation may be considered homogeneous by reckoning x as of one dimension, y of n dimensions, and

, dog , consequently of (n − 1) dimensions, and of (n − 2) dimensions. In such cases assume y = x" U, p = x»-lt, q = quin-2 v; then by steps similar to those in the last case we arrive at a differential equation of the first order, between t and u, which being integrated will enable us to determine the relation between X and y.

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The other factor +- 2u = 0, gives

y = CX, as a singular solution.

dy dy If x be reckoned of o dimensions so that y,

Y, da' dx are of the same dimensions, a homogeneous equation may be integrated by assuming

p = uy, 9 = vy; whence as
dy = uydx and udy + ydu = vydx,

dy - ude and

= udx and du + udx = vdw.

From this last if we eliminate v by means of the given equation, we have to find u in terms of x, by integrating an

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(24) Let xy dx = y dx + x (ax) + ( a x):
The integral is
(ai? – x;)- blog ¢{nb + (a” – xo)}}

ny b and c being arbitrary constants.

V. Equations of the second order in which one or other of the variables is wanting. If the deficient variable be the dependent variable y, by

” we have an equation of the first order between p and 2, by the integration of which we obtain p in terms of r, or v in terms of p; and then by means of the equation

y = Spdx = xp - fxdp, we can find the relation between x and y.

dy) na d'y
(25) Let

+ 1
*dx! ) * 2x dari

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putting = p and

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c and c' being arbitrary constants.

If the independent variable (x) be wanting, we put dy dy dp dp

, and then we have an equation bedx? dx 'dy dy' tween pand y from which by integration we find p in terms of y, or y in terms of p, and then x is known from the equation

dy = pdx.

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whence by eliminating p we obtain a relation between and y.

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