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- duz +QUR
But u, is a particular integral of (32) without its second member; so that
*+Q2Ug = 0. Again, let na be another particular integral of (38); and let ú, be another particular integral of (32) without its second member; then, pursuing the same process as before,
so that ug and u, are two particular integrals of (41): and employing Ug as above, it will be seen that
so that ug / Uzdx is a particular integral of (41); let this be equal to u'z; so that , = uzsuzd«,
. dm3 - d ua d ax in ... Uz = dx u = ax
(44) a na
so that now U7, U2, Ug are expressed in terms of ni, ng, ng, that is, in terms of three particular integrals of the given equation, when its right-hand member vanishes; and these may be substituted in (35); and the final value of y thus obtained will be d ng
d ? ad ne de la dx ni delodnes y = n) dx ñ at ax d no the Jag hax n dx d na dan |
đa 1 ) The same process may manifestly be extended to equations of the order n; the final result however is of a form too complicated to be inserted : it will however involve n signs of integration, and therefore n arbitrary constants.
444.] Some examples of the above process are subjoined. Let us first consider the linear equation of the first order, viz. a + Py = x.
(46) Now of this equation, when x = 0, an integral may be found as follows:
dy + pdx = 0,
y = ce-Spdt,
(47) which is n; and therefore substituting this value in the generalized form of (45), we have
y = e-fpds (xeSpdt dx ; and which is the general integral as before expressed in equation (69), Art. 382. For a second example, let us consider
heya-bay + 11 a2 din bay = eme; (49) particular integrals of the left-hand member of which are, when the right-band member vanishes,
11 = ear, na = elat, n3 = 2342;
= e* ser des sea de serem-se de
= (m—a) (m—2a) (m—3a) + 2' 23 28 48 + 2242 +cz els ; (50) and this is the general integral of (49); (49) in fact having been deduced from it by the elimination of C1, C2, and cg. Another example is
y = x; the particular integrals of which without the second member are n = e*, na = e-*; and the general integral is
y = Cyc* +0,e-*-X. 445.] The process which has been explained and illustrated above also gives the following Theorems.
THEOREM II.-If m particular integrals of a linear differential equation of the nth order without the second member are
known, the integration of the equation with the second member will depend on the integration of a new linear equation of the (n-m)th order.
Let ni, ng, ... Nm be in particular integrals of (26), when the right-hand member vanishes; and let us, in Art. 443, assume y = /vndx.
(51) Then substituting as in that Article, the coefficient of / 0, dx as exhibited in (28) vanishes; and we have dn-lo, dn-30, dn-37,
(52) dan-i Tei dxn-1 + R2 domn-+ ... + Qn_19 =
mi Now of this equation, without its second member, according to the process pursued in Art. 443, (m— 1) particular integrals are - d na d na d nm.
(53) đx ni dx n Tx n' let these severally be symbolized by $1,52)... Sm–1; then in (52) let o = $i /v,dx ;
(54) and substituting according to Art. 443, the term involving 10,d« will vanish, and we shall have
+ R, Zone3 + ... + Rp-2V2 = dat-2
(55) dx-3 " **-2°2 ,Ğ and of this equation again without its second member (m-2) particular integrals are d 6 d $z d Sm-1 ;
(56) dx Ġ' dx' & '* dx Š which we may conveniently symbolize by 04, 0g, ... 0m-9; and by a similar process we may make the integral of (55) without its second member dependent on the integration of an equation of the (n-3)th order : and in a continuance of the process it is manifest that each of the given particular integrals of (26) enables us to reduce by unity the order of the differential equation; and finally therefore the order of the equation will be the (n-m)th.
Hence, if a particular integral of a linear differential equation of the nth order can be found, the order of the equation may be depressed by unity.
446.] THEOREM III. If nu, Na, ... Nn are n particular integrals of a linear differential equation of the nth order without the
second member, and if y, is a particular integral of it with the second member, then the general integrals of the equation with and without the second member are respectively y = cını + C2 12 + ... + Cm Ym+yu,
(57) y = C1 Ni + C2 12 + ... +ennym The truth of the proposition is evident from the form of the equations; because each satisfies its corresponding differential equation, and each contains n constants: these however must be independent of each other; and the particular integrals must also be independent of each other : for suppose that n3 =anı + b ne, then
y = (1 91 +62 12+cz (an, +672) + C4 N4 + ...
= (Cz+acz) +(C2+bcz) N2 + C414 + ... and which contains only n-1 arbitrary constants, and consequently is not the complete and general integral.
447.] M. Libri has in the Memoir above referred to traced many analogies between the formation and properties of algebraical and differential equations : some of these are given in the following Theorems.
THEOREM IV.-A differential equation, linear in at least the first two terms, may be transformed into another linear equation of the same order, and without the second term.
Let (26) be the typical equation of a linear equation of the nth order : and let y = uv,
(58) where u and v are two undetermined functions of x: then, ex
. dhy dn-ly pressing and wony, ...... by means of Leibnitz's theorem, (26) after substitution will become
d" u dv dn-lu n(n-1) d2 v dn-a u
the second term of (59) vanishes ; and from (60) we have
(61) whence, theoretically at least, v may be found; and (59) will be a linear equation without the second term.
And more generally: A differential equation of which the first m+1 terms are linear may be transformed into another linear equation of the same order, and without the (m+ 1)th term, by means of the solution of a linear equation of the mth order. Thus, let it be required to deprive of its second term the equation dy _3a dy +2a®y = x;
(62) substituting y = uv, we have
let 2 - 3av = 0; so that op een then (63) becomes de manu = xe *
448.] THEOREM V.-If a relation is given between two particular integrals of a linear differential equation of the nth order, the order of the equation may be diminished by unity:
Let n and ng be two particular integrals of (26); and let us suppose them to be related by the equation ng = 0(,1); then, if in (26) we substitute for y, first ni, and then na, or, which is equivalent, $(nı), there will be two equations from which
d.rs may be eliminated, and the order of the resulting equation will be only the (n-1)th. Thus suppose n and ng to be two particular integrals of
-ay=0; and suppose them to be related by the condition na na=1; then we have