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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 403r, page id 00068196:0801
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crescente X crescil y
Gy dx + 2yXdx = XX dX + agy dys
yy + CD 2 XAXy + Xde
sit ex = - x + () et dy =
 Ege  medx Lyt nd_ = X  (Et etre du
 dertu, sis teutus lexitus
dx-ily duelus
aa Xdx Fry dy elaaxay +aydx L 2a a Xdus
(2)ox. fiet 2dy. dx. 29 ddly +6yy ys tay llx + adx dy
215 dyt ay dxqedy- dx=
=225 y + 4y dx 2 ly. x 3, =, 4 dus - 4 dyax + ex  44  C LA aa dx
y dx ddus
(3
seu (4y ay day) + 12 y du + 2aydy lax (+ 2a dxay   tgly - 6. y. dxdi agersa . a tu tu
(= 4 is dyadys =cay dxddys + ry ty tax (+ ay Ax d X) = (4 aa dxalus  liysiae r aidtpo
12l-Esaxtay 4 29 y dex = 0 seu fiet 6d3yda dius t gdus t d
6dnd (5)
6ci - yde ddue 1 G dy ddx = eg et in oy. restabit 2ytys dax (+ 2 duaus  y dx daxf ax L
=2ydX dayue (F y dx dax = (D dxds = (4) dy dx Y dx
ubi si substituatur ex aeq. 6 valor ipsius y in aeq. 5 prodibit aequatio
ita igitur inveniendo dy pery, adeoque et ddu pr oe dex
equatid
habemus potest in aeg. 5. tolli du et des diu, adeoque ibi
remanebit solum aequatio qualis oporturtt inter  du, ddus
Unde sequitur posse pro arbibrio
regulaniter) assumi divelli aequaioes differentio- differen
qualem fingere licet.
tialis modo id fiat cum fructu, ut hoc loco
1
e
in pte est quoties e quo
Tae Methai ad diferentiali est solum
quantitas determinata est sotu unius gradus, ut hoc loco aa
tantum in ea datur
(1)
si sit y-42
a fint dy- ardo- avdx
n
us cn liceat
Ca)
extractione dfferentiando fet aXda i a x + 29 dy 469 di t a ylix  a dydeo
 fiet a e ax = 2ydis: dx = aest et a d  - 2 2dXy ddy - 2dx 3y dy + 2y dyditu
2)
-atx ydye seu a + 2aydu + 3du =y du du 2 dydiu El y x du ara ax
seu a=  yVy de ay- 213 i daque -69 ax ays + ryd cy dxtax
et ex aeq 3. fiet sg seu fiet seini eoplicando a ( nempe BB
403
ex
ex a faciendo a in advalor. et ex 2a faciendo a + avabr.) fi et
Eydyax  y de dis etc.. Tfa de ydy = - 2dx y lgs - 6dx  ti +2 y dus dax
ydy dx = ydy sas a us .. ..
quam aequationem divellendo in duas. habebuntur duoe
aequatines differentio- differentiales, earumque ope tollendo y
habebitur unica aeq. inter du, adus, de ddx.  solubilis.
tle. Necesse autem erit ex his investigare quaenam isto casu
sit quantitas constans, ut in
Car cum ex a videndum an non tollem
aliam
duo
ex et dax
verent ne incommoda producat
aliant polee qan op  inguloren
o duas
tetotats  y,
videndum an non cum dantur aequationes differen
tio- differentio differentiales, licett aequavem divelles
in tres. et pro altioribus in plures Et idem est si plures literas indeterminatas adhibeam
407
u e e spert aeit aniterie  aest teribut