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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 97r, page id 00068196:0191
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Sit lata aequatio differentialis 100 +10C + 120 Coco + 10ideo = (1) =0
Assumatur a =() 217 +20,: 317 +30, et explicetur aequ. 1. 7er 2. fiet
+109. 30. 30 +, 100. 30.31 + 100. 31. 31 yy
+110. 20.30 + 110. 20. 3Cy
116. 2, 30y + 110. 21. 31 yy
+120. 20. 20 +2. 120. 20. 21 y + 120, 21. 21 9y
+ 101 d2o. 30 4 101. d+ 30y
+101. d20. 31y + 101 d21. 319y
= 101. 20. d30 -101. 20. d314
E
= 101. 21 d30y = 101. 21 d31 yy
Hic patet non posse tolli terminum absolutum, ubi ab est y et du; nec terminum
qui habet yy. quin incidatur in aequationem a similem datae,
nam _ res reducitur ad 31E 30: 20 quantitatem quaerendam in termino absoluto
et ad 31: 21 quaerendam in Termino qui habet yy. sed nec potestterminus dus
tolli, fiere eni 31: 21 = 30: 20, quo facto co fieret quantitas constans.
D
Unus ergo qui tolli possit est terminus y. Et  ad hoc sufficit esse 20 =0 =31
seu esse co = 217: 30, et esse 3 constantem, fiet enim 110. 21. 30 + 101421. 30 =0 seu
fiet d21: 21 = = 110: 101.
Effingamus aequationem differentialem summabilem, et
videamus an huic identificari possit. sit 9 + do = (4) =- et sit
 V (5L s dl + ydm + mdy H yydp + 2pyd ( log ((+ my + pry) Ergo
L + my + pyu
explicando aequ. 4 per 5. adifferentiaque aequ. 5, et explicatam 4 ordinando
fiet 1), +gl = gmy +gpyy
l
(d
by Mheyys Hedus +hfydy =0
identificabimus ergo aequ. 7, summabilem, seu ex summabili factam cum aequ generali
multiplicata per h,et
pareseu articulum 9 _ identificationeque
patet secundum du fieri
instituamus. Identificando terminos dis utrobique, fiet m= e se m = (10) = he
secundum ydy fieri p = hf p = (11) = hf: 2. secundum gy fit g =  hc - Lhaf - L. fdh, :bs
seu g =  (12) = 2hc - hdf - fdk, : hf. Secundum y fit g = (13) = hb - hde- edh,:he
quos duos ipsius g valores aequando, evanescet utrobique dh: h, dividique poterit utraque pars perhi
 nam fiet ihc - hdf,: hf (= ah: h) = hb -hde, ihe (-dh: h), adeoque tantepr
prodibit 2cif - df. f  = (14)  b: e - de: e. t re t exeds atquod si abesset.
b et c, uti sane semper abesse possunt, oporteret esse inf: C  ration constan
Ansic B = (5) = (_ + gamt mdy + plt pnyt gllus totmoes
l+ my
ds g + + gl + gmy
set +armus
alatus + matyus+ lfdue + Mtlus
pl +omus
+ 9l dy +9m ydy
+mdy
r9 , et fint tritus
ficrta etm g= ltf - ene +fm, :, m. l
n h = tmm -Eml Altf-F me + fm, : (n - 1)f
 hat hby s hegy+ hedus +hf ydus
fic a c. f = dt: t +9
poragus
Hic, p, g, h carent differentialibus, ergo ab is incipiatur gy est m = hc: dt = F+ 4 hf:, t 4)
dl + dmys
et fiet et + c9 = fdt, seu q = f dF - Ct; C, godus, firce lt + (fat tet + chI
ob dy sit 1+ + 1f dt = lct + m H hE = dC hC = hC, seu ficet taB. + lfalat lc taC) tech = chedt
habetur F
ubi c ponatur abesse
Exsolute, et m perh
juperest y, et y. sane ar est superfluum, possent etiamq et de  se destruere seu poni ar = -g imo et
9 + dr + p, possunt haberi pro una quantitate X et super sunt ob y, et y duae aequationes
H = - dl: 1 + ha: l = lat: te = (Bde Be -Bdh: C) + hb. de ubi non licet ponere rationem inter
et si constantem, evanesieret enim sit ergo _l = hst, fint dl = hdA +Adh, et aequatro talis prodibit
-T.): 3 AAB. F (dh: H) + 4 :d = A dt: e - de: e (= dh: h) + b: e. sed ita incidimus, in aequationem
datae non cedentem
Ergo ponamus b adesse + abesse C et b, firc at = 0
opeat nn im aleadta eo spreus a 3  btm. (i 1000 dor a an buer 3) d hiaet cas
h = tm + am,: f
+ 101. 21.3
-101. 21