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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 97r, page id 00068199:0193
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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