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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 308v, page id 00068196:0612
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sit data aequatio differentialis intuscunque gradus. Assumatur aequatio diffecetralis
progima altior genaralis, talis ex qua inta per simplicem differentiationem
deducta poni possiti et haec differentietur.  Ex hac disperontiatione et data tollatur
quod opus est, ut urialur aequatio comparunda assumtae
Eso data disterenlialis  100 HBy 410. ay t eogu Au ydus = 0 Assumatur aequati
H _ 200 + 310y + 220 yy =0 L cuius differentintur est d200 + dvo y. + 20 dy + 42oyy + 2. 120ydy =0
Etipagus. et 3 fint - dy = 10 +Oy +1 20yy, :, 101 HIIy = deo, + duoy + dvogy ,:9 80 + 2. 220y
Et multiplicando in crucem 1902 ∞t H0 20G + 120. 21079 habetur aequ. in qua erit y
y, y, y
Ergo aequatid assumta, multiplicanda esset per arbitruriam simplicm. Sed vellem ego
evitare ductum  assumtorum numerorum inter se.
C1)
data 100 + 10y +Bol dys + 120yG+1 11 dys = 0
C.
Assumater Diferentialis alia A nubitraria 200 + 20Y + 201 dy + 2oyy- + 21 ydus = 0
An ergo sic
Harum duarum aegu. ope tollatur du, habebitur aequatio ascendems ad
. y +. y + . 9 +,. 9 =0. Hae differentietur habebitur vq (1), a altior utique
quam 1 vel 2, cum ascendat ad yd ut gydus.  Ergo multiplocetimip e per
ioquo nn hil atrabinus tuibus umini pastenteris aequatiouis 2 et perante erit ac si vito  d
formntu arbitrariam 300 + 200y fiet aeg. (5 eut acs adi
t s di sed nine destirsre aita
tet aqB. Sed nunc lescybient plure arbitrariae
constantem per constantem et habebitur aeq. 6, idinificabilis ipsi 4. Ita merae habe-
buntur a Ita aequationes identificatoriae carebunt ductu incognitarum
inter se invicem porre aeq. 4 habebit  terminos  7. artitiarias
autem determinandas assumsimus etiam 7 nempe 200, 20, 201, 220, 2, 300, 310,
primum autem tollemus 300. 3i0, quod per facile, restabunt 494. 5. differentiales
simplices, adeoque soluriles, in fallor per methodum paginae praecedentis
Ay
de
aa
11)
assumatur 200 + 2g0y + 201 dy + 220yy + 2ny dus 20. Ex1 est dy = 1 100 4 120793 : 101
Ex 2 A d -dy = 200 + 210y + 220973:, 20c + My, et multiplicando per triucem
fiet - 1o1. 200 = 10l. 20y + 101. 220yy
compendiscibatur
=100. 201 = 100. 211 y - 120. 201. = 4 20. 21 y3 7 =0
300 + 3909 + 320yy + 330y =0 Et video revera aeq2. nm
gydy,
frisse opus lifferentiamdo fint 4300 + d 310y + grody +dhro yyr 2. grohdys + d 3o + 3. 210  
multiplicetur sequ. 1 per qu + a1y, fiet 6 1on. 40 + 1 40100 + 40. coldys + 40. 120yy
+41. 100y
440.120y
Huic addatur mallip ceg. 2 multiplicata p si opus (t per 50 +G17, sed ibi 50, at Si
erunt constantis, sed fortesse suffecert multiplicari per S1y, quia, et primam
reperiantur 4O et 4I. Hinc jam opus est  explicari 300, 310, 320, 330
sed per gradus, non enim epus est statim explicari, sed proat opus videbitur
ubi scilicet litea quaeritur quae latet, sufficit explicari eam in qua latet
Et hac methodo non video quo obstet quominus quaesitu consequamur.
Operae pretum erit calculum institui et absolvi generaliter pro data
diferentiali secundi gradus 100 + 1407 + 10. dys + 1207y +102 dy =0
 as tioio