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where Pn, Pm-1, ... Pg, P1, P, Q are functions of x and y.
If this equation is integrable once without any specific relation between x and y, it must satisfy (13); and consequently
de d2 p.
-...(-)" = 0. (18)
dxu If it is integrable twice, it must also satisfy the condition
. By continents ... (-)- Ti = 0; (19) and so on. Thus if P, is a function of x only, the equation dy, dp,
y = 0 satisfies (18), and is evidently integrable immediately.
Again, if P2, P1, P, are functions of x only, it is required to determine the value of P, in the equation
I d2y. dy
(20) so that the equation should be integrable once; in this case (18) becomes
is an e
is an exact differential expression ; and of it the integral is
per a + (2, - ded)y=c; (22) suppose again that (20) is integrable twice; then in addition to (21) we must have from (19) P, 24= 0;
(23) and this condition might also have been deduced from (22), by applying to it the criterion (18), that (22) should be integrable once.
There is also one other point that deserves notice. Suppose that (20) does not satisfy (18), but can be made to do so by the introduction of a factor; let u be the factor, then we have
- d2y dy
so that (18) becomes
(25) - dx T dx2 and if from this any value of , general or particular, can be found, then (20) may be integrated directly. It will be observed however that (25) is a differential equation of the second order in terms of U, and that the difficulty of solution, as far as the order is concerned, is not lessened.
SECTION 2.- Investigation of Properties of Linear Differential
Equations. 440.] As there is no general method of solving differential expressions of the second and higher orders, we are obliged to have recourse to such particular forms of them as have yielded to the powers of analysis; and amongst these the most remarkable is that known by the name of the linear equation, and of which the solution is of the form y = f (x); into which the independent variable and its derived-function enter in only the first degree, and where the coefficients are functions of the vari. able & only. Thus the most general form is ny dn-10
+ Pny = x, (26) 1 dir-i teen where P1, P2, ... Pri x are functions of x only. Of this equation we shall prove some general properties, and then proceed to the solution of particular examples.
It will be observed that two forms of this equation have already been integrated ; (1) in Art. 150, where P, = P, = ... = P., = 0; and thus
(2) the general linear equation of the first order in Art. 382, viz.
441.] THEOREM I.*--The integral of (26) depends on the integral of the left-hand member of the equation; that is, on the integral of the equation when x = 0.
* The first of the following theorems is due to Lagrange: the others are the original investigations of M. G. Libri, and are taken from Crelle's Journal, Vol. X, page 185.
Let y = un/v, dx, where U, and v, are two undetermined functions of x : then by Leibnitz's Theorem
dm-20, du, dm-10. ...tm
dxTM-2 da t dm-* u; (27) and substituting the specific values of this in the several terms of (26) we have
n-1918 = x; (28) where Q1, Q2, ... Qn-1 are determinate functions of x and Uy. Suppose now Uy to be a function of x which makes the left-hand member of (26) to vanish; that is, suppose u, to be a particular integral of (26), when x = 0; then the coefficient of /vndx in (28) vanishes, and we have
den- + Q1 2n+ ... +Qn-ematen + 2n–10 = (29) which is an equation of the same form as (26), and of the (n— 1)th order; in this equation let
v; = uzsv,dx, and let substitutions be made in (29) according to the preceding process : then if ug is an integral of (29), when the right-hand member is equal to zero, the resulting equation will be of the (n-2)th order, and of the form
dr-2v. dn-3 v.
Öxn-3? +Raca-+ ... +Rm+3 deze + Ru_gos = m ; then continuing the same process we shall finally have an equation of the first order which may be integrated by the methods of the preceding Chapter; and the function which satisfies the given equation will be determined by the successive integration of a multiple integral of the nth order. The problem will hereby become reduced to that of a multiple integral, and of simple quadrature.
442.] And to indicate more clearly the form which by this
process the last integral assumes, let us consider the case of a differential equation of the third order,
d2y dy .
+ Pi Lenz F P2 dort
Now if u is, according to our supposition, an integral of (30) when x = 0, the first term of (31) vanishes: also let
Again, let u, be an integral of this equation without its second member : and let
Again, let ug be a particular integral of (33) without its second member; and let
and retracing our steps we have
(35) where Uy, U7, Ug are integrals of the several equations found as above and without their second members; and thus the general integral is found in terms of a triple integral whose elementfunction contains one variable; and therefore by the process of integration three arbitrary constants will be introduced, and the integral will be in its most general form. And thus to generalize the process, the integral of (26) will be
I U Ug ... Un 443.] Now these quantities U7, U2, ... Un may be expressed in terms of particular integrals of (26), when x=0; so that if these latter quantities can be determined, the complete solution of the given differential equation will depend on only a single integral. To limit the extent of the investigation, let us confine our attention to an equation of the third order, viz. diy dạydy
+ P3y = x; dx2 TF1 dx2 + P2 dr
(37) and let nu, Ne, n3 be three particular integrals of this equation, when x = 0, so that
den + Pro + P, 92+ Pgni = 0; (38) then employing Uy, U2, U3, V1, V2, v, in the same signification as in the preceding Article, let uy =nı; and if for n, in (38) un/undx is substituted, it will on expansion be seen that
so that wy / Uzdx is a particular integral of (37) when x = 0; suppose this integral to be Ng; then
:: U, = dx ni
PRICE, VOL. II.