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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 296r, page id 00068196:0587
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14. jul. 1691
sangentes inversa, partem aequationis
differenbialis summando, et reliquum hansformando,
reducendo item ad ee homoeoptotas
ydx: dys = 292bXX + CX 6yx -2bX +Cequatio differentialis solvenda, contenta
in literis Bodenianis jun. 91. Multiplicando per crucem, fiet:
6xyydx = 2bXydx + cc ydx () 293 dy D- bxx dy + ccX dy. dividatur
aequatio per yu fiet: 6Xdx b L b XX dyjgu cCx dy Fydx. grs
pap
b Xd
seu  12XX  bXX: y + CCx: y = 4 gy. se Bxx
seu 4y2 - 12XXI + bXX = cCX Z0 non y7 - 3Xxy + bxx = CCX = 0 ut sreral
habet Epistola Bodeniana.
Sydx -xdy an etiam dabit X. a y
Respondeo non dabit, nam d X auu adx ydx 1 e2C 
2
nam d, X a-y = a dx - ydx= xdys). Interim hinc patet sempeu
seu homoeptotas esse X et y quoties formula diferentialis ess summa
bilis, excepto quoe interdum, cure  si quidem sit  formmula in integris.
si vero accedante erecti ad fractum quid ae accedat,  tunc solis
signis possunt differre im Si noluissemus adhibers in aequ. Z ad hibere divisionem
per y, videtur res etiam debere procedere. Adoxjydx respundet
(656XX ydy sed ita nihil luciamur. Itaque res semper ita instrituenda, ut
plura praesentia si licet simul summemus. Quod dende resituum est, tactemus ut
quod si rem semper reducamus ad homoeptota, faciendo
X = _IB Z1 ∞V FIX,B y = 30Z +40V +3HI, ita ut coincidant 10, 20, Nem e
licet transformando.
et ex aeq. t fiet yi  30307 42. 30. 40 ZVC.  2.50. 34Z + 341. 341
+ 3 403Z
40.40 
sit ergo formula xgydx jam gy = 30 3027 t 2. 30 40ZV + E. 3031Z + 34134
C
N = 10Z + 10V + 11
2
40.4udv
1 = 20Z+ 20V + 21.
fiet Xy = 10. 20ZZ  10. 20ZV + 10. 31)Z + 11. 21
11. 203
4 40 341Z
sit formuta pro possita
XX = 10. 10 +E + 2. 10,10Z P + 2. 10. 11Z + 1111
yg = 20120 (7 + 2 20 20 ZV7 240217 +2121
N’y = 15. 202 +3. 102022Y +210. 20HXBZZ + 2. 1020. 11 Z B +, 10. 21. 217
ZoV, 10. 1021)D BV +210. 10, 21
2. 20, 2612
X = 10 Z + 3. 1823V + 3. 10. 117Z + 3. 10. 11.11Z + 113
vo
Zev
Prrro si detur X e X dy + y, ee
Quaeramus differentialis fomularum homoeoptotarum
b de du dv L Lvav d v + 7 = dy + dZ
d vo + ZZ = 2vdv + 2Z43
dv + Z3 = 3V do + 3Z 43
dv + 9+ = 4v’ av + 4 Zd3
dezue = vdZ + Zdv
(D∞D 
+ 11. 11. 21
20. 11. 11S V
Et  y, yx
eodem modo scrabuntus
mutatis rescondentibus
dCdZZV + ZVV = Z2dv+ vvdZe + 2)B d3
C, dx
dv3Z+ eZ v = vAz+ Z + 3 Z vdZtsvrzar di
uP
210
id
Et ita porro pergendo; hinc patet rebus  redu reductis ad homve optota
issima reduci qua difficultutem summandi tantum enim opus est
nam in ommi formula rationali differentiali sculmandee hymocoptota
sutumanda
summanda qu’integra differantialem habente. non assurgentem ultra
dimplices, et nam quotrescunque dx: du habetur rationaliter; 
tep et non assurgitur ultra gradum Z2 tantum opus esst, ut z vet vn
ut coefficiens ipsius _ sit duplus coefficientis ipsius Z20 et vd 
et quotiescunque non assurgitur ultra gradum 73, tantum opus est
coefficiens ipsius Z praeter conditionem praecedentem, ut coefficiens ipsius
E ita porro.
Et o stab Leo or Hs tipeitis cosiccales iptus 2 a.