456 
Divide by @246y, and put z x instead of y, and the 
equation 
dy te: : d«=0; 
instead of dy put now zd 24d, and the equation is 
transformed to es et 
—* (a+bz)dz - 
bP + GFE th 
The fluents of the two parts of this equation may now 
easily be found. As a case, let yd y + 
(«+2 y)dz=0; then, because a=0, 6=/=1, g=2, we 
move <2 =0. This is easily transform- 
=. 
e tig¢e7 GFR 
and hence, taking the fluents, 
Led h Ob) + pee 
or L fe(r+29} =-y and putting y for xz, 
1 fe(e+y)} +575=° 
Ex. 2. Let ay™dy+(2"+)y")dz=0; hence we 
have 
1+é:- 2d 
dy+ tt= d2=0, and a gia ead =0. 
lace the tustehle uantities being separated, the fluent 
maty bo foal te waat: . 
FLUXIONS. 
innit tent fanctions of z alone. This equa- 
= called finer, bt “opened gency gre, Coke 
pms yor equation of and of the 
order. Paneregi fn, substitution the equa~ 
tion becomes 
cdtptdeqPatda=Qde; 
ody? make an prec inet 
os ial the coefficient of 2 oan =0;, is 
AEF eet d mn ( 
The first equation gives =- Pda, ‘and. aes 
L (Q=—f Pde; and as Pax does 9, its 
a Ap td ; peti oskirbeer: 
fluent w 
then aL te L@® "capa Lied, 
quantity) and ¢ =e ~*** =e=" es Aches 
@'Ws the numbier of whieli Bap: deal Syed og = 
for the constant quantity ¢ " We now substitute 
value of ¢ in the pee Ly leap yh ced hoe 
A dz= Qe'ds,and hence’ bogus 
Az=J Qe" dake " 
edyprrieis pore ri a) 
* ar get RED aloe 
* and at last we find. ae Ati: ona 
olusde mT. 
ye “=f Qe" aon o Wiaioaia rPas. 13 
From this we may infer that; ‘was not ne- 
cessary to add the constant quantity a to the fluent 
Ex. 3. Let sdy—yda=dx4/(2?4y*); we divideby fPdz=u, as it ag appsered sanin i Sia mabecanet 
x, and have dy— 2 da=day(1+ £): And ma- 
king y=x 2, so that dy=zd2+42dz, we have | - 
dx_ dz 
z Va+e) 
hence (art. 128.) z=ez+4c,/(1 + 2), or 22=c 9+ 
ev )» wide, by tran: PE Oe 
a2 cy+er 
“F170. In some cases an equation may be rendered bo: £¥ 
mogeneous by transformation. Thus, in the equation 
loatbotediys(net-ngtn)d =o 
we € (ar+0y7+0e)=2, mz+ny+p=t, 
hence SOS8S aged wd epee, 
and dy= mdz—adt : oe bdt—ndz 
en b—na’” — imbue 
The proposed equation now becomes <d y+tdr=0, 
or (mz—nt)dz+4+(bi—az)dt=0, which is homoge- 
If m 6—n a=0, this transformation fails; but then, 
na 
™ =~,» and the proposed equation is 
bedy+bpdz4(az+by) (6d y4ndz)=0, 
the variable quantities ed pails separated by ma- *s 
dz—adax- 
king «2 +-by=z, by which dy= org The equa- 
(e+z2)dz 
~ ae—bp-+ (a—n)a 
171. Let us now consider the equation 
dy+Pydz=Qdz, 
part of the calculus. 
EXxaMPLe. Let the equation be. ay 4 gdu tends; 
then P= 1, Q=aH,u=/Pde=a, ' 
SQe*de = fests e*dz= ae’ (2®—82?+6%—6), 
therefore y =ce  +a(23—32? + 62—6). 
172. early analysts classed fluxional equations 
by the number of their terms. In such as of 
shee seems, ape wine Sheneiane heck the fen ant * ae 
pees ie terms camprchenel nth 
yu ‘Fas pue pe du at ° oldu. stihy 
‘This may be put under a more simple form by divid- 
ing all the terms by 3 «2%, it then becomes 
Be Se hes ate 2 
ere Alga ss ena at 
a el a Se B cna ir to 
(Kft 1)2 Ra yah 
19+ @—itily! ’ 
Eton _ (kf tide at ef 
re ang wg te a 
een 
- a 
. 
~ eek 
