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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 7r, page id 00068199:0013
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10 yy +19+ n =0 sit aequalio data Y dyzdx = 10 + ny+ ergy r erc
+e
20f 219722 9y
et sit assumatur alia dy. dx = 30+ 3ly+ 329y,, 40+ 41y+ 42gy fetc,
tellendo fiet aequatio in qua du: dx fiet aequatio in qua incognitae invicem
non ducentur, hac differentietur, et inde ducta dy: dx, naequetur
datae, habebitur rusus, aequatio secundum X et y, quam aequando pricti
elucidentiando habebimus meras aequationes simplices differaitiales
primi gradus une ducetur quaesitum.
si aequatio sit differentio differentiatis videtur eadem sufficere
assume et assumendo
via, nam faciendo Xdax- axd
d=ddy dux =0 assumerdus a day _: dxdy = 30 + 31y + et
40+41y
et ita toltendo dius habebimue aequationem in qua solum X, y, dus, dx, quam
rursus differentiabimus, et pursus ex cognita ope tollemus dy, et habebim
conferendo prion. aequationes simplices coincidentiales, potuisseri
s
eliam assumere amplius ddy: dadu = ex X non tolum per dysedpere
sed et per dya ita plures aderunt arbitranae.
ad
si aequationes Si differentiae ascendant ad quadrata et subos
vel alias potentias, deprimentur ad diff gradum differentiatem
proxime inferiorem; ulbi prodibit rationalis vator, et deinde
inscentemus  eadem methodo reductionem ejus ad proxime
sufieriorem, quam ita assequemur bis, et ita habebimque quaesitum
Ita quod notadum nunquam atertitiae ancoguitae
despendent ultra primum des differentrationis gradum.
21X + 22 dZ + 20
Guix + Brray + 10 -.
piny + pady + 2Z + 22dZ + 10n. A = 0 ~ py 411~y + 12 2+, 127Z +100
l+20
et 21= 121 et 22= 122
a
fil nn = M et. _ p = 112
11-y + 12Z 4 10 10  21 2 + vrd +20= 0 Expredle s H yt ply + 12 Z + 1+d10
patae sunt duae aeqq.
et
111y + 112 dys 4 1213+ 122d7 + 100  0
211 y + 212 dy 4 22174222d) 4200
tolutur + 11I. 222 y + 112. 222 dy + 121. 222 Z (+ 122. 212 dzu) + 100 202
=211. 122 =212, 122 = 221. 121 = 222. 122.
assumatur adhuc una aequatio
= III. 222 My + 121. 222R
=2. 1222 n  = 22. 122R
= 112. 222M
= 200 122
et tollerodo dus fiet:
my+ndy Y p =0
+ 100. 222n
=. 200, 122 H
= 11Z. 222 p
.Haec aequatio differentietur
inde ope datarum tollatur Z et dz
fint aequatio coincidentinda assumtae
4 212, 122 p
Alia via incidit melior, et plures exhibens arbitrarias datae suntaquationes
4212. 122 M
(ny + 112 dy 4 1217 + 122dZ +100=0 ft assumatur tertia 31y+ 31dy + 32DZ + 322d3 + 300 0
ubi omnes oefficientes sunt arbitrari
et 2H1y + 212 dys A221, +212d9 +200l. 20
ope aequ. L2  1at 2 tolla inveniantur dus et dZ, hi valoris substituantur in aeq. 3, habebitur vg. 4
hous solum Z et y. cujus differentiatio erit aeq. 5. jam misus valores du et desubstituantur in aque
dibit _q. 6 secundum Z et y coincidentianda aequationi 5, in hanc autem rem adhibe:
tco prodibuit aequationes 3, quarum alama p quae pro ultimo est termino
p300 d er a38 non videtur labitur e dificultatem; at suae fonerunt aeq.
quibus 311. 312, 321, 322 et horum differentiae: Hae aequationes ernt simplices, explicemu itaque
si mul tolli v. g. possit 31 et d 311, qd ob multitudinem varm possibile credo, ha reducta res
ad unam aequationem, nes adhuc literas haventes quae patbituo divellor
 facit
 mulpes et habebitur vaelu. Hol