Page images
PDF

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

[ocr errors]
[ocr errors]
[ocr errors]

N = 2 (m12 whenever m=,_ ", which is one of the conditions determined in Art. 424. And if n is a whole negative number, then

-4n n = 2(m+2); :: m = 2n1j; and this is the other condition found in that Article. Hence 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

f"(x) = f {fn-1(x), fn-2 (x)}.

[ocr errors][ocr errors][ocr errors]

and the equation is a differential equation of the second order; of this let the integral bez = $(x);

:: f(x) = /**(x) dx»–2;

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.

Ex. 1. avetiye me i la line = y-b. . where b is an undetermined constant.

i nte att det int log (y=0) = sq; where c is another undetermined constant. Ex. 2. d y = dx (dx2 + dy2), where x is equicrescent. d y

î = dx ; . i. log (dx? + dya):

[ocr errors]

(dr.

[ocr errors]
[ocr errors]

Or we may integrate as follows: the equation is

[ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][subsumed][ocr errors][ocr errors][ocr errors][merged small][ocr errors]
[ocr errors][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors]
[ocr errors]

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

[ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors]

Multiplying both sides by 2 dy, a deling - 2017

... = 4(0)-c = lyd=243;

All equations of the forma y = f(y) may be integrated by this process; for multiplying both sides by 2 dy,

2 dielny = 2 dyf(y);

i rayon = 2 [f(y)dy +c; 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

[merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][subsumed][merged small][merged small][ocr errors]
[ocr errors]
[subsumed][ocr errors][subsumed][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors]

log (p+a) = log contentc; but as ay is a transcendental function of x, the next integration cannot be effected.

[ocr errors]
« PreviousContinue »