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# Catalogue record: 370 — [Aufz. zu Differentialgleichungen]
# Incipit: Datae Aeqq. 100 + 111y + 112dy + 121z + 122dz = (1) 0
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=370&katnr=57946
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 97 (from the shelfmark LH 35, 13, 2c Bl. 97): 2 pages, canvases 193–194, page ids 00068199:0193 to 00068199:0194
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=193
# Lines: 126 recognised on 2 of 2 pages, mean confidence 0.85
# 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.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 97r — 00068199:0193
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000193.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 72 lines, status machine
atae Aequs, 100 + 111 y = 12 dy F 121Z + 122 dz =0
200 + 211y+212dy + 2217+ 222 d 2d
  400 + 4IIy + 412dy +4214 + 422d7 =0. uti
Hubita aequ habetur aej. B) adeoque et Aqu. 4.
+ =11aZ= 0
ubi pono 321, et 312 20, sic enim  qe omponeriu tentemus ipsasi et
fortasse videnmus in simplicioribus Multiplicemus dats pe f et g, atque inde componamus,
itaque fingamus haberii ipsa (3 et 4ratus
quaeslaum primam 3, +300f +315y +312f dy + 3 etf 1 + 3. 2fdZ  100, t u( H12dy +1217 +122 4Z
odem modo rarsus multiplicemus dales per 4 400g. + A1gy + Hgdy FarI9f quigd2 2 + E y + 2 a tus 221 BZ + 222n VZ
i _ 422g = nin trun 42g = iim t2itn Ergo (um tian) l = (121p + 229) 9
O
h,l; et quarllas per pia 5
42l = 12 p + 2249
Et (122m + 222n) L = (122p +5229) g fit umtrum = 4
Eodemmodo 422 b = 122 p + 2229
Superest 311f + 4119 = 111m + 2uIn
122p + 22c
312ftang = am + 2Nn
12em +22In
121p + 2219
h = Mprrq. Nul,; 31 = 12p t 2ng. trilis 3
312h +412l = u2p +220
f= ium + ZIIn = 4119,: 31 = 12m +212n = 41293: 372
3uIh + 4ul = 111 P 4 2110
Tandem 300f +4009= 100m + 200n
Et 300h +40ol = 100p + 2004
fit f = 100 m +20on-4009, f. 300 l = 100p + 200q 4vol, . 300
prioribus fit III. 300 M + 211. 300n- aug 300. 41g = 100. 31m + 200. 311n = 400. 349
W’alor postremus ipsus f aequetur dugbus
112. 300m + Z11 300n = 412, 3009 = 100. 312m +200. 312n - 400. 3129
= 100. 3up + 200. 319 - 4u0 3Il
Et 112. 30op + 212. 3004 = 412. 300 l = 100. 31 2 + 200. 312 9 - 4u0. 312l
Er eotem moddo ex fit 1. 300p + 211. 3009 411. 30
Hatemus ergo aequationes 10, nempe 50, 2, 3, 7, et 0, D, 8, 6,4, E
M, au, si Gto 4, n 4n an 31. 312
supersunt 300, 400
item Diorae ex m, 2, P, 4
literas 41, 412, etC.
4u = u1 m + 2uIn = 3u f, : 9 = uIp + 219 - 3 uhr, l sit 3u = nulm + 2U Im Hegg 2u94u i, f -ti
sed malium f, diferri, quaeriqus adeo aer tris isins L valores et h
pro
e412 = 112m + 212n - 3125, : 9 = H2 + 212g - 31h, : l fit 312 = 112lm + 212ln - 112 9p - 21299, :, f. - h,
400 = 100m + 200n = 300f, : g = 100p + 2009 - 300h,: l, et 300 = 100 m + 20479 3:, f. - h
Hac methodo videntur omnes aequationes disferentiales simplices
primi gradus quotcunque incognitarum solvi posse, et quidem per
interventionem merarum ordinariarum aequationum
si sit aequatio duarum incognitarum praeter unam varientem samtam pro cognita
ita ut liceat tassumere novam aequationem videamus quomudo inptitui possit
assumtio v. g. 100 + jHy + 112 dy + 1 dy + 121Z + 122 dz =0 hoc catures
lacillima est, nam fiet 12 dus 2. hoc casu facilis est divulsio fit enim
  200tmyt 1r dy = 0 et ,  + 1212+ 122d2 = 0 et 300 + 200 = 100
ubi alterutrum 200 vel 300 est arbihrarium prorsus.
sed si sit 100 + Icy+ 12 dy + 1+ Z, .. ydy
Eliaf videtur brevior via esse haec, ope a94. dataum 1, et 2, tollantur quaerantur
valores ipsarum aus et dz,  per meras cognitas, hi substituantur in aeq. 3, et +
lum quae vantur valor ipsius y, et valer ipsius Z, et uterque sed uterque
fial identicus, ita ut y et Z evanescant ita qualuor juo quia utrisue
valoris communis divisor, tres aequationes destrnctibiae sa tit facient quaesisu
quod unrum, et incogntae non vescendunt ad differentias. Hoc pulcheriinum
Et locum habet ubicunque pes aequationes differentiales pluriliterae sunt
i plicari hoc potest ad collationes ordinarias in differentialibus
selverdae
investigundis, ubi res credit ad auxiliares sunt rnerae differentiales
simplices, sed ex sed mistae incognilis ex mullipleate oit
Ex. gu. sint aa sit aequ. diffialis du: Ax = kx 149:aa, assumatur
10  + uy 4 129y+139+ 14 20 ubi quae aequalio differentretur, et tollendo dus: dx
ope priorbus uartur, nova aequatio (3) e, itaque multipricetur data per
tomNg. 2. per formulam arbihariam, ut comparam possit aequasioni,
aequationes comparalitiae continebunt simplices aequ. ubi tantur sunt
y
10,  1 18, 14, ara tus discaertat, talentur oiffeata

## Folio 97v — 00068199:0194
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000194.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 54 lines, status machine
ope harum aecquationum restabunt simplices 10, 1314, esc.
ce
stabunt et simplices, quamqalviram valores, et destruantur
fingantur lolinem novae sem simplices, quot est opus ad tollendas
10, 11, 12, 13, 14, Earum ope tollantur insae  lio, dir etc.
restabunt 10, 11, 12, 13, (4, hoorum quaerantur gatores, qui erunt
reddende identici seu aubiles ope tam assumtam iterarm
arbitiariam in fictitis seo equationibus quam ope ipsam
aqua arbtiariam ad mulhiplicationem assumiatum
 cum scilicet multiplicaretur aequad 2
Ilya
100 + 110y + 101 dy + inory H e y  aequatio data resorvenda ast
sumatu 2e0 + rioy + 3oidg cujque tam habetur polutio metiplicetur per
Asumatur ny et fid gede +  +ny +ndus, hoc multiplicetur erfy
ul y an
fiet  (B gm +  m + fny + fndy + gnydy + gngy =0
hoc aequetur datae fiat ntd dueo
+ funu
agmys +gndu
et gn = I1I, et fn = 101 et gm = 110 et fn = 100
habetur n costanti
+gangus
f = 101 :n ergo un m: n = 110 seu m = nOn : W1 et 101 m: n = 100
sed ita res nimium determinatur F sit n+ dn -0, erit n constans ut dixi
y = CH : n
et aequalio ert fn + comus +fndy + gnusdus ~ 100p = 10 sy + 101 pdys + II py dy
fiet p = fx p = gn: 11I Ex defit fn = 101 gn : 111 Ergo fg7 g = IU1f: 101 et p- (1d n fu.) 101
dy
Ex yfit am = Uop seu IIfm : 10I = 110Fn : 101 eu _ m: n = 116: 11, superest
fp = 100B seu (n = 100F1: 101 sud ita fieret 100 = 101, quae praeparatu an sit
postibilis videndum sit 200 +210+ 201 do + 11IVdV, fiet v = fx+1 et
11. 200
210. I1 20 dfy
duc (dy ds. V + fdys + du
ergo prostat collere dis 20183 pe C99. fm + 9my + fn dy + gnydy =0
11d1
et o 100 +10fy + 101dy + 119 dy fit dus = fm +gms, :, fn + gny sed ita
mulplicatio evanescit. Ergo resumatur in + (1 + du) y = - dy = 110 410y, : , 101 + Iy
sit 10im +  1 my. + 11I (1+ du) gy C = 100 + 110g ie non procerit
+(1+ du) 1017
si elbet m Allumatur in yy + ny + p et  sit in constans
et fiat mgu + ny + + p + ny Q + mgy + gndus f29m ydy C7 00 Aicoy + rat 12099 + 11 dy F 111yd
qap tgany tam gy E
qu = 101C gm = WII. 2A fit 9 = 101: n X IZ: 1
et m : n = 2. 101 : 111
seu fit E = 2m, : 2. 120m uam et 9 = 110t - n, : du = 101 t. n. seu 1otn - nn = 10lF dn
rursus q=. 12 c = m, :dm = 1nF: 2m et fit 2. 120t 2 m = 11 tm
et nusus t = nM, :, non - 101 du, ubi pro quia  n = 2. 101 m: 117 = Bm,
2.1om - iIdm 11oBm - 101d Bm - 101 Bam, ubi B cognita ita videtur plena haberi
Asmmi
liet mm
porro p haberi pd pro 0
solutur, quia feliciter evanescit nm umo reperitur am: in fieri=
52egu