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and, the limits of integration being (x, ) (x, y), if (2) as well as all its derived-functions remain finite and continuous for all values of the variables within the limits, then (2) can be integrated in a series, by the method of Art. 367: and its general integral will contain n arbitrary constants.
Also it is evident that a differential expression such as (1) may admit of integration by reason of the form of the expression, and independently of any specific relation between x and y: the conditions that this should be the case have received much consideration from Euler, Lagrange, Lexell, Poisson; and lastly from M. J. Bertrand*, and M.J. Binet, as quoted in Moigno's Calcul Integral, Vol. II. p. 551 : and it is to Euler and to the last two that we are indebted for most of our knowledge of the subject. In the following articles the conditions requisite for such a case are investigated by means of the Calculus of Variations.
Suppose the integral of (1) to be definite, and the limits of integration to be those particular values of the variables which carry the subscripts 0 and 1: and let the definite integral be expressed according to the notation of Art. 247. Now our object is to determine conditions which (1) must satisfy, so as to be the x-derived function of some other function of the form, du dog do-ly)
(3) dix' dw2" "dxn-1) independently of any relation between x and y; that is, so that
[ r(x, y, Ý.... y") dx = [*(x, y, y, ...yn-1] (4) and so that this equation may subsist independently of the functional connexion of x and y.
Suppose this functional relation to undergo a small variation, and the values of the variables and of the (n-1) derived functions at the limits not to change; then by reason of (4) the value of the integral will not be altered, and therefore 0./ P(x, y, y', y",... y(m)) dx = 0;
(5) then employing the notation introduced in Article 303, it is manifest, that if we replace the left-hand member of (5) by its
* See Journal de l'Ecole Royale Polytechnique, Cahier 28, Paris 1841, p. 349.
value given in equation (56) of Art. 303, (5) cannot be true unless dy' (PY" d" y(n)
(6) 1- dit dm2 -... (-)" on = 0; and this therefore is the condition requisite that (1) should be an exact differential independently of any relation between y and x.
It will be observed that y, Y', Y" ... are partial derived functions; but that the subsequent x-differentiations are made on the supposition that all these quantities are implicit functions of x: and that they do not vanish, although x may not enter explicitly into them.
435.] Let us now pass to the converse of the above. Suppose that F(x,y,y',... y(n)) satisfies the condition (6); then its integral is capable of being expressed in the form (4), and independently of any relation between x and y; or what is tantamount, if (6) is satisfied, the integral can be expressed in terms of the limiting values of the variables and of their derived functions; and this is what we mean by definite integration. For in this case, by virtue of equation (56), Art. 303, the variation of the integral on the left-hand side of (5) will be expressed in terms of the limiting values of the variables and of their derived-functions; and in terms of these alone; and consequently the integral will be a function of these quantities only. Hence also, if these limits are fixed, their variations disappear, and the variation of the definite integral also vanishes. Some examples are subjoined.
Ex. 1. Let v be a function of x and y: it is required to determine the condition that v dx should be integrable independently of any relation between x and y.
In this case (6) becomes Y = = 0; consequently v must not contain y.
Ex. 2. Under what circumstances is (P+Qy')dx, where p and Q are functions of x and y, integrable, independently of any relation between x and y? In this case (6) becomes
which is the same condition as (28), Art. 371; hence also we may infer that the complete integral of the differential equation of the first order and degree contains an undetermined functional symbol
Ex. 3. Prove that y'y" — xay y' – xy2 = 0 satisfies the condition of integrability.
436.] It is good also to exhibit a posteriori the criterion given in (6) in a particular case. Let us suppose yy to be the integral of a given differential expression, when no functional relation is given between x and y; then the x-differential of Yy is
941" + - = F(,y,y,g"); and this must satisfy (6); now
437.] We may also by a similar process determine the conditions that r dx" should be integrable m times successively, and independently of any particular relation between x and y; m being not greater than n which is the index of the highest derived-function contained in F. Let v = F(x, y, y', ... y(");
(8) then it is manifest by the principles enuntiated above that, in accordance with the notation of Art. 150, the variation of the definite integral of
V "P(x, y, y',... y(m) dæm must not involve terms containing signs of integration. Now using the symbols of Art. 303, and supposing ôx = 0,
and therefore the right-hand member of (9) consists of a series of terms of which (12) is the type; and wherein k receives all integral values from k = 0 to k = n, both inclusive; and where Yo = y.
Now it is evident that, if 8. / "vdxm is to be free from terms under signs of integration, the coefficients of dy under the several signs
| must vanish of themselves; whence we have dy d2"
This series of conditions must be continued so long as the integration-signs have positive indices; for when the indices are negative, and when they vanish, the corresponding terms have their limiting values : of the general form (12) therefore we must take the last m terms; that is, the terms corresponding
to values of the indices of the integration-signs until k = m-1; in which case we have
dy(m) m (m+l) d2 y(m+1) y(m-1) m t
+1 m (in +l)...n da-m+1 y(*) ...(-)"-m+
1.2... (n-m+1) dx"-m so that we have m equations of condition; and if these are satisfied the given differential expression will be integrable m times successively.
438.] A similar process enables us to determine the conditions necessary that
F(x, y, y', 5", ...y("), z, z',z", ... Z()) dx, (15) in which we have used the notation of Art. 308, should be integrable independently of any relation between x, y, and z: for if the variation of the integral of (15) does not contain a quantity under the sign of integration and depends only on the limiting values of the variable quantities, then
day(n) dx + dxz
and similar conditions must be fulfilled if the element-function contains any number of variables; and also conditions similar to (13) and (14), if such an element-function is capable of m successive integrations: thus suppose vdx" to involve m variables besides x, then the number of conditions requisite that v dx" should be integrable n times successively is min.
It is beyond the scope of our work to investigate the corresponding condition in the case of a multiple integral: the student, however, desirous of pursuing the inquiry will obtain the necessary aid from Jellett's Calculus of Variations, and from Moigno et Lindelöf, Calcul des Variations, referred to in the foot-note of page 411.
439.] Of a particular form of differential equations of the nth order, which is called the linear, many properties will be investigated in the following sections; but it is convenient to consider it at once in reference to the conditions (13) and (14). The equation is