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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 301r, page id 00068196:0597
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possumus eriam assumere, ut Z non sit dy. dx sed ait aliquid pralica
ut 1nA + aX + by = CX dycdx + C y dy: ax e sit Z= ex t ey dus: ax
ser dy =   _ 2dx: Cx+ ey et. ed fiet h + ax + by Z Z adeaque fiet
a dx + b dy = aZ et per 3. fit x a Cx + ey dx + b7dx = ext ey dque
pam
ex 0q. 4 est y - Z. b a h. b = a X b. Ergo sit a cx + a Z. b dx - aCh: Cax + a CX: b x
+ b7 dx = CX aZ + a CZ. bdq = a eh: C du + a a ex: b dque sed uit hinc lucram
Tale quid in mentem verit sit aequata, vg. h + ax = ex dyndx t Cy ay: du
h + ax + bg = CX dy: dx + eydy: dx seu ax + b9 2. ex dyidx + eydys: dx h
fiat 7 =, Cx dyodx + eydiy: dx = h,: X. fiet ax + by = XZ, adeoque
E sit seu a+ bgia = Z seu a + bD = Z posito v = y: X
a X + by  =ZX seu adx + bdu = ZdX + X +Z
a x abus (xy + Z fit adx + bdys = xay 1B dx + d2
seu ex aeg. 1. est y = ax = Z, : X - b ergo fit a dx + bilus, x - b = X dus-
a ex eb du    dq  = b - _ dus X. X- b = aX - Z +2
aX _ Cd ax C, Z Z) + Z sit a p = Zdy 17 dZ + dq
ax aby  C+ 1 xy sit a    E ZXy
aial
ax  by = ZX7  CZ_ - a X + CZ - b y 20 L2) mX+ ny
fit X = 6 = ny: m   D HL -m ndus - m ydZ +   ny de, = mon dx
ubi proes substituendo et fit y umdx  madus.
et fit imax + mndus, ndque - maque = y
et un dy + mnde,: maz- ndz = X
et atbos valores substituendo in aeq. 2. siet
mundus + mmndx = mmndx- mundus
et ia tollantur moest evanescit et de
sit mx + ny = h sit X = - ny: m + h: m et axmni
et de nim ax  = mndus = my dque + ny dque = hdZ
Ergo y.= um ax + undus +hdZ,: n. - m aZ
Et siciliter X = nm dy + nn dx + hdz : m =n dZ
mundus +mun dx + nhaque = mmndx -. mundus - ulidque = tmaque - hndZ
Ergo luis valores subitituendo in aej. 1. fiet
quae rursus est identica. itaque nihil sic Cucramur, nec possumus collere simul velfis
itaque aluid notificium at me utem venit, ubi praesens artificium ponendi
A  Z = dy. du comb et quaerendi 1y. combino cum alio attificio
 observatione, qua deprehendi semper posse aequationem esolvi quando differentialer
1od
per se solae servent lomogenertateur.
rno ad set aliqua constans vel plures primum
solvi, quando X et y.
sem per plures constantes
pro b poncae possum Ba
et X sint numeri sola vero
edicemus ad no uinam sint enim a b icu
et pro c ponere possimus X a. ala ab k sit β
a sit linea. t Ergol egc. sit ergo 2 xi Et sit
et sit aeq. (27 proposita, inter X. y. Z. a
Huijus quorantur differentiales duae, (3) et (4); quas combinando cum ipsa 2, habebimus
sit ergo BCC du = Zdx.
duas oquationes (5) et (6) in quibus aberit a, jam ope aequationum 5 et b tollatur, sit a
g et ope aq 1 tollatu  et ope aeque 1 tollatur. dus. habebitur aq (9) in qua
et x adcoque
erant solum Z X de dx. servantibus legen homigenectum. es
solibilis erit per quadraturas. sed quo verenda destinctiares restreipsa
Tetnda sit L, Z te s L CxZ + C yZ fit hi CxZ
 Zdx = dy es  h = exZ + eyZ ubi h es constans quae solu turbat homogenersolem namus z est
ratio, et c atque e, sunt ut cumeri R fil CxdZ + CZdX + Cy dZ + C Zdy = 0 quae est sine R.
varsus h: Z = X_ + ey. fit 2dZ.  22 = c dx + edu -  - ex - ey, Z. Z
habenus ergo duas aequatiores in quibus abest a in quibus tollendo du per aeg. 1. sit
exaeg. 3 CX dZ F C Z dx + ey aque + eZ2 ax =0 et ex aeg. 6 fit ZCdX + CZ2d X. + Cx d2 + ey dq = 0
30
30.1