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# Catalogue record: 285 — [Aufz. zu Differentialgleichungen]
# Incipit: Sit aequatio data dy : dx = [10 + 11y + 12yy + etc]/[20 + 21y + 22yy
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=285&katnr=57860
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 7 (from the shelfmark LH 35, 13, 2c Bl. 7.92): 2 pages, canvases 13–14, page ids 00068199:0013 to 00068199:0014
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=13
# Lines: 222 recognised on 2 of 2 pages, mean confidence 0.86
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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## Folio 7r — 00068199:0013
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000013.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 71 lines, status machine
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

## Folio 7v — 00068199:0014
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000014.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 151 lines, status machine
sint ergotres aequationes 111y + 12 dy + 1L1Z + 122 dZ + 100 =0
Companto i1 dy +2 dz + 101 -0
Compendio
21 dy 4 22 dZ +20 L0
2119F212dy + 2213+222d2420020
ubiduae pudus aequs sunt datae
tera assumta
Tollenda du et dz
aZ = 10. 21 - 11. 20, :0 11.22 = 12. 2 et dy = 10. 22-12.267:9 12. 21 = H. 22, quae duo mutatis
31AG + 32 dZ +30=0
signis habent eundem denormoatorem. Ergo ex aeq. 3 fiet: 10. 22. 31 = 1. 20. 31 -. 10. 2(. 32. 4 11. 20,32 F12. 43 1 2c 3
fiat LIII. 222. 31Yy A 12L 121 222311Z +100. 222. 311a Jum 1223
D(- 21I. 122. 311 9  - 221. 122. 3117
O=  = III. 211. 322y
+ 211. III.3249
+ 121. 211. 311y
= III. 222.3119
= 121. 211. 5227
= 221. III.3227
+ 121. 211. 3228
= 200. 122. 31 a
= 100. 2II. 322a
+ 200. 111. 32a
+ 121,211, 300a
= 111. 222.3272 = III. 222.300a
411  311 y + 421. 3117
412 3227
422917
per 
dix (P.31y
je.. y.
(6 322y
I1.22 - 12, 21 =
111. 222= 1224
+ 431 3ua =0 fit quae aequatio
432. 322 a
439. 300a
diferentiata
dabit d41. 2u y + 411. 311 dy 4 d421. 311Z + 421. 311 dz +14 31. 31 =c
4I1. d3H y
d412. 322 Y
412 d322 y
412. 322 dy
421. d311X 422. 322 42
Hanc aequationem possemus appellare
a422 311
422 d3117
compendio SI y +512 dy + 521Z + 522dz + 500 =0. Eamus
431 d31
4432. 322
432 d322
d430. 300
430 d300
conjugedo quam cum aegq. 1 et 2. inde facere aequationem, qualis O, et cum ea coincidentiare
sed quia taman postea ob nimias, arbitrarias instituenda divulsio est, destmerim
prius dus et du. sud veres hoc quidem si liceret. Sed si liceret si adderemus hun
aliquam ex datis vel summam datarum, ad conservandam legem justitiae
Et ita videtur res processura, sed video ipsam  malecalculatam et omissas
sed multiplicat
esse 8312, 321.
prius pene arbit
Et tum assumamus relationes inter binas quide,
transcendentales simplices, et binas alias ordinarias simplices pro effi
crendis qualuor aequationibus. Et ex ordinariis assumlis quaesita, tenta
scirvabunque s trauscendentales suas simplices. Et ita tentemus efficere quae
situm.
s
bona
sed nu imnia fabricanda est nova aequatio C
4102L + 1II. 222. 311y
= 122, 2I1
+ _ III. 211. 512y
k 11I. 211. 312y
+ 1II. 222. 321Z
= 122, 211
+ 1II. 222. 300a)(+11. 222 fAy +11. 322 321Z +III. 222300a
= 122. 211
-4 121. 211. 32Z
=127211
=122. 211
L - 122. 21)
+B100. 211. 382 a) = 111. 222.312y -121. 238.3137 = 100. 222. 312a
4g 11I. 221. 31Z =41II. 20. 3(2a + 122.211.312)+ 122. 221. 311Z + 122. 200.312 a
 +4I1I. 222.312y -121. 222.321Z
_100. 222. 37a
42
+ III. 211. 3229+ 121. 211. 322Z +700.211. 322 a
= 1H1. 211. 392y = 111. 221. 3222 = 111, 200. 322a
12 dgo- 112 dy + 123 dy + 1. fiet =d7 = 112 dy +109: 122 Ergo dy = = p10. 312 + 929 )20 pr du - t0 25 2. 120
4 L F122. 2V1. 312
4 122
0- 212 dy 4 222 d +a
o 312. l9 + 321 B t30_
AE
4112. 222 3Iy +H2. 222.3212
=122. 212
= 122,212
-HI. 232. 3121) = 121. 272.312Z
+122. 211 312y 4 122.221. 312Z
4 III. 21 322y 4 121. 2]. 3322
= 11. 211 3329 = 11X371. 372Z
= 212 dy + ∞10, : 222
2. 222- p2. 21
quos valores substituendo in aequatione 3tia seu initio assumta fit
= 112: 222. 300a
= 122, 212
= 100. 222 312 a
 122. 200. 312a
+ 100. 21. 322a
= 112, 200.3224
compendio
+1222, 311y 4 1222321Z + 1222 300a
= 1122. 312y D- 21223127 -0022. 313a
+11I. 322y + 2112322Z +0012. 322a
quam d
=1222 d311 y
 + 1222d321
L +1222 d300 a
=4122. 312y = 1122. 312dy = d21 22. 312 -2122. 3Bcy =d0022.312a
= 1122 d 312 7
= 2172 d312
2 = 0022 d 312a
+d1112. 322y + 1112.322dy +2112. 322 +2112.3224Z +-d0012, 322a
=(112 d322 y
praestat ergo tollert Z112, 432
dz v
20
quam aequationem differentiando
fietc: +d 1222. 31H + P1322. 311 dy + d1222. 321 Z 4 1222.321d7 41222. 300a
D-
4 0012 d3220
quodsi haic aequationi addas datas
binas arbitraris multiplicatas
lice bit deprmere dus et de, et
ia regia per eorum valors, uti numeri non pisi cogniti. et per set
habebumus binas aequationes, in quarum qualibet sunt 31. 312. 32l. 332. odeamus, an 312 et 3EZ sic explicari liceat, ut sotum
in una aequatione restet 311, in altera solum 32I, et habebitur quaesitum. Itaque hoc jam examinandim in genere
sunt binae aequationes differentiales simplices primi gradu, tinarum quatuor literatrum  variantium, vidiamus an hacu
harum binas sic interpretari liceat, ut  in una equatione una lterariliquirum, in altea altera restet quo
dus ( - 102 a. myn defly = 1e2 + idZ fiet y= cps BSudx ((ax, watrczfiredq = W0 ((et dx, 100 + 2 Z e de
posile spes erit
Agn. 2ec + 2119 4 2417 1222 dZ hrty = l sit bsitu b.  =  Adx, 0d00 + 22i Zt 2224
ubi sin sent datia
Syt 312 dy + 321∞ + 322d3 4 300 20
fiet 12dZ+10, : 11 = 22 4Z F 20, : 21 seu 12adZ + 10, 21 Z U. 22 dZ + 11. 20. Ergo
quod superest coincidentiare psi
aequationi C, sed ita spere
duae aequatione pro coincideo
tiandis Z y et Z aderunt duo
aequationes quarum quaelibat
habet amb incognitas prorsus