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Now these integrals admit of integration in finite terms whenever n is a positive whole number: and since, see Art. 425,

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which is one of the conditions determined in Art. 424. n is a whole negative number, then

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And if

Hence

and this is the other condition found in that Article. arises a reason why Riccati's equation can be integrated for these values of m.

The reader desirous of further information on the integration of linear differential equations by means of definite integrals is referred to the large work of Petzval on the subject; viz. Integration der Linearen Differentialgleichungen, von Joseph Petzval, Wien, 1853.

SECTION 5.-Integration of certain Differential Equations of higher Orders and Degrees.

462.] As no general theory exists for the integration of differential equations of all orders and degrees, we are obliged to have recourse to artifices, which analysts have from time to time devised, for the integration of particular examples; I propose therefore to examine the most useful of these processes as concisely as possible and in order. And firstly I shall take differential equations of higher orders, where the highest derived function is a function of either the one next, or the two next, inferior to it.

Let f (x) be the highest derived function; then the problem is, to discover the integral of the equation

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and the equation is a differential equation of the second order;

of this let the integral be z = 4(x);

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so that the final value of f(x) depends on a function of x which is to be integrated (n-2) times in succession. Some examples are subjoined.

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1

2

( dx

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y—b = = {e2 ̄a+e ̄(x−a)}.

Or we may integrate as follows: the equation is

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= x -α ;

=

{1

dy2

d2y

1+

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It will be observed that in the former of the two methods we have integrated first with respect to r, and in the latter first with respect to y. The final integral also might have been found by

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Let

= 2;

... az

=

(1 + z2)$;

PRICE, VOL. II.

4 P

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= (x — c) { (x − c )2 — a2 } 1 — a3log (x − c + {(x − c )2 — a2 } 1) + C12

whence y may be found by integration.

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which is linear of the second order, and with constant coefficients.

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(y1 — k1)3 (y1 + 2k3)

=

All equations of the form

d2y
dx2

= f(y) may be integrated by

this process; for multiplying both sides by 2 dy,

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and of this the root is to be extracted, and a subsequent integration is to be effected.

Ex. 8. Determine the curve whose curvature is constant.
Let the radius of curvature = c; so that

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463.] Let us also examine differential equations of the second

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where c and k are the arbitrary constants of integration.

Ex. 3. a2 d2y (a2 + x2) + a2 dx dy = x2 dx2.

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which is a linear equation of the first order in terms of

therefore may be integrated.

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Ex. 5. dx3 dy-x ds2 d2 y = a dx ds {(d2x)2+(d2) 2}3,

where s is the equicrescent variable.

ds2 = dx2 + dy2, 0 = dx d2x + dy d3y ;

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is a transcendental function of x, the next integration

cannot be effected.

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