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# Catalogue record: 60595 — [Aufz. zu Differentialgleichungen]
# Incipit: Cum ni fallor solvi possunt aequationes differentiales simplices
# Date: 1706
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60595&katnr=57497
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 282–283 (from the shelfmark LH 35, 13, 1 Bl. 282-283): 4 pages, canvases 559–562, page ids 00068196:0559 to 00068196:0562
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=559
# Lines: 278 recognised on 4 of 4 pages, mean confidence 0.83
# 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 282r — 00068196:0559
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 283v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000559.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 71 lines, status machine
cum in fallor solvi possint aequationes differentiales simplices, id est, ubi
nulla incognitae affectio in ulteram ducitur, tCogitavi porro
quanam ratione liceat efficere, ut ope aequaliones comparatiaae proalus
aequatimbus solvendis non aliae sint, quam simplices
Esto aequatoo data 120 gy + 11 4 dy = 110 ay + 101 a dy + 100 aa = ()  = 0
e ssumatur 2207y+ 21ydy + 210ay + 201 a dy + 200 aa = 2 =0 , ubi evefficientes
arbitrarae Follatur du, fiet 120. 21y + 110. 211 agy + 100. 211 aay
120. 20) agys 10201 aay + 100. 20l4 2=0
11. 220y +1I. 210 ayy + 11I. 200 day + 101. 200a3 d
fujus aequationis 3, quaeratur aequatromo
101. 220 ays 101. 210 day
Muloplicetur aqualio 1 per (310y + 30 dy Y 300, et assurgetur ad aeg. (6)
en
Hormulam
quae coincidentienda est aequationi 4. Habemus lite ocoincidentiationes septem
et aequa, literas assumtas arbitrarias quinque ex aeq. 2, et tres ex formula 5
id est acto quod sufficit ita enim rei reducta erit ad aequationes
tem
Lhocmudo nori
vidtamus an generaliter pro quocunque gradu sufficient adsant
simplices, differentiabs quatuor
arbitrariae sit aequ. 1. g +. g. dy 4. y 4. y ay. etc. =c
fiuent
termini 2C+1. Assumatur aequatio 2da, quae dabit corbitrarias eto
similis, quae dabit arbitrarias etiam 2e+1.
Ex conjuncta utraque fit
aequata 3tia fist dy = Y +..,:,: ge +.. qui valor habebitur bis, eosue
valores Emjungundo prodibit aequatio 3 assurgens ad y
habebitur aequalio 4, in qua erit
sunt 40 -2 + 1 = 4e -1.
2-1
Haec differentietur
ren . Gre dy +... Cujus termini
n
affollenda est aequatio , quae erat y. y du
V
Saaee
-1 reedus etc, ubi arbitrariae sunt e 2+1 = 2
ut fic q. ,
Fprum am (5) esrd t
quas adde Et prodibit 69 16) comeidentiantue aequationi 4, ubi coincidentiationes
eam
quod fiet multi plicando per (V
sunt +C -1 arbitrariae autem ex aeg. 1. erant 2e+1
et ex formula5 sunt ete
reti cum tamen solum indigea
Mus arbitrariis indigeamus arbitrariis 4e -1. Sic ergo res non semper procedit
quas addendo prodibit de prodibunt arbitrariae 
Itaque assumamus ubi opus est aq. 2. altiorem, sit ergo Aequ. t)
noy +u du + etc =0 Aiquatio assumta sit ) 220y + 211. y du t etc., fiet
dy = 120y +.. 9 , - 111 ge1 et dy - 220 3 +..,:, 21 . Unde habebimus
aeq3)
etn.1  geanar = 0, quam differentiando habebimus aequationent)
+.. yern- du +.. = 0 Cujus sunt Termini ret 2n -2 + 1 = 2e+ 2
u
itaque aeq. 1. debet multiplicari per  Formulam 6y .. ist, en.
1
5)a
porro
ut prodeat aequ (b.) comparabilis aequationi 4. jam aeq. 4 habet habet
in qua
terminos 2e +2n -2 + 1 = 2e+ 2n - 1 Videamus jam quot habeamus arbi
et (formula 5 sunt n, fiet 3n+ 1
et debet numerus arbitrariarum 3n+1. non esse minor numero terminorum
trias, eae su ex aeq. 2 sunt 2n +1
coincidentiandorum, seu + 2e + 2n - 1, seu n + 1 debet esse non minor quam 26
seu a n debet esse non minor quam 2e-2. Atque ita semper hac Methodo
si aequalio data contineat dydus, vel du, dyt, etc fortasse optimum erit pro
sufficiens erit numerus arbitrariarum.
evitanta exaltatione arbitraniarum descendere ad dy ddu ita alia itidem tali
assumta tolletur ddus

## Folio 282v — 00068196:0560
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 283r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000560.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 89 lines, status machine
differentiales
pus est ergo ut aequatines simplices exslideruus solvi posse
doceamus. at si sint unius incognirae, erunt y +. dy + y. =
y + ady +. . dly 1 g. = 0 et ita porro. Has
f = 100 fieret 11-
pro ila C   109 , 1 ty + 12 ddus = 100 essume tq y + Zdy
+ Av dys + vddus
av
1
12
10t = 1 y + rtli us
fint 10D = 128Z et (Z+ dv = 4. 10  seu) Z+ du = n 0. Has duas aequationes ut solvamus
1Z - - dlv + H do - vd 
= 10 v si esset 1u - 
Wd = v: 12 = Zt du, : 1 = dZ: 10
fiat Z - _v. a et dque =  d8v + fdv:a, et habemus duas aequationes
012
1e B  adg V + 12G do , et b_B + adv - 12 V = Fiat Z =  pv + aG, : a sit 10 = D1. fiet 4   - B
fiat du = p D dp + pdB + a , : a Ergo 10aV = 12 dp V + 1 psv + 12 adg et uv + agt Mado = 1 a V
sa V et 12 p do + 12 a d de = 10 av - 12d p - 12 a dg,: 12p = 11 av - 12 pB - 12 a: 12a sed
sit y - _WV, fet ex 10) + Udus 4 12dadus
seu i(t ita non licet tolea v, quia ascendatur ad pP +p
Aliter si sit Z = pa + dg: a v = quo  + d t9
Fot nolpio + 11N aH +n Vadio
11du W + 12dip d1o
12 da  s
1096 + 10 + 8 = 12p deo + 12 co dp + Y d e + n6 dr
22Z 12pcd + 12rC + 12 gdco + n2codq + 12d18+ 12Fdg = 19uo + ut 9
(e 4 lr - 10 +t) B
fis t = = βY. ut evanescat a
multiplicata aeqque. 1 et 2 addantur in utum fiet
in nr -utg (127 + 1dt- ut s a
et ut oua nes cat e fit 1Er + uBY4. 12a βdr - 12 ardB +  12aβdr + 10BB
12t a + 12 (3Y, d8 = 0
ergo nonlicet destrure simul  et des sed videamus an liceat destrine 8, et ded
non sunt simul
rec et B asiendit ad B(
de, evanescit ri
seu g= - pB: a. sed
quia ile fieret D= (3Z, et ita non nisi die proorieat
129dc + 12Pβdco =0
sed ita destructo
 destrncta peraq NV. _ proco, fit p  129 12β ap  1039 +. 12 ap + 1radg.- H a a
tollere simul  e  et
ob deo, fiet
permussi hi duo valores psius 9 et ipsius E
Ergo oportot
ponamus p= 0, habetur g ex aeq. Z et pourmus + =6, habetur l ex aeq.
librae capitates, sed una.
Reassumaiuus ergo. sunt duae aequatines 100 = 12 42 et 2Z + 12 dv = 11 B
(it V = qco et 22 rg fiet 12rdg + ndr = 109e0 = 0 et 1erG+ 12 gdco + 12codque - uqcd = 
et addendo in unum fiet has duas aequationes puore multiplicata per β, posteriore per a fit
(20  ut ergo tollantur  et co, fiet
11a + 10()
d= - a et da=
fiet do de 2 q: BY, inventis q et Y, B manente arbibraria
+ 12 ar + 12dq( +12a9 dco
= 119 aco
e
si
+ 12β3r de
+ 12Bdr  = 10β910
in constans foret, seu ipsa Y. Quod hic non licet
et seu deo = 12r 9 - 10 B9c0; 1249, sed
sed ex oqe6 fit deo = +ne taga - 12rG, : 129, quis duos valores aequando fit
quamnon licet facere constantem, alioqui et
et ergo (212 a91
+12 β dr G.
+12 a Y 8
12d9c0 sed hoc inutile, quia nihil aliud prodit, quam
= 10 (3 9 W = 0
= 1a 9 co
+ nadu 1o
quod assumsimus ut _, et co eranescant, alioqui
potuissemus haec rursus differentiare. vel elia,
ld tollere ex aeg. 5.
+co + 1t8 - 12qp dcd
Hac ergo nimia sunt
+11900 + 11 +8 = 12 9deo
-12b..
12Y
A superiora reundi redenndo d_ = -12d  - idrd a - = - 12dg. - 12 at.
liet itaque ut tollamus ded, fint pt = Yq seu + = rq: p et Z = pC F r∞, :a, et v = qco + 148
seu Z= Pco + re
et v = P9co + Y48 sed ita fiereb v = Z9: p, quod est contra intentum
Si utrobique tolleremus et 8 firiel. 10tt - ntdr = urt- nrr- urdt quae aequatio
Ubi quia assumi potestC in ratione determiinata ad Y, habeti iun rar
atque adeo quositum sit Y r fiet (ren 1) Y (10+ 12 + 11) er 12 rde seu axir = cotur cer. 12
i

## Folio 283r — 00068196:0561
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 282v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000561.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 70 lines, status machine
Reassumamus ergo
o= dzue y + Zdy
U data sit equatio coy + ady + 12ddus = 100, assumatur
L
100
d dy + vddy S  (2
02) 10Wy = 11 lodys + 13 loddy = 100)
10 = 43. 10 6) eb Zr du, : 11 Z v: 12
fint 10V  12 dZ et 1Z + ndvZ uV Has duas
aequationes, ut solvamus assumo q c0 + B8 v= 10 + =g
Ex B fiet, 12dao + 128 dr + 12 rdg = 10c0 + 10+ 9 et ex 9 fit 12co t 1XG + 1ndeo + 12td9, 2 d 110 4 1t
10+8 - 12adeo (15)
ido urq- iadco Ergo ut tollamus _, sit
 = (per 13) + 1l0 = 1 at
1010H - 12 tar = -12rr. 12rd L. ur1
sit t = Pr: a. fiet: 1a Ber - 12 a βrdr = ncarr - 12 aBrer 2 β + 11 a B YY uhisi F r fit (1o  HY=
14C
pondum solutam
=12dr 
Et patet
ite idem prodibit, si pro t: r substituatur B, quod est in potestate. Itaque exitus hac via
= 12a rr dB
sed video nihil agi et evanescere r, et d quari per ae quationem
An sic procedemus  10y + 1 dy +12 ddy = 100
non habetur
tantum Ergo opus est ut faciamus 10y + adue summabile
su
fial y = 8V et fiat 10DB + 119do
11 dgV +c do - c d
Sit - 108 + n d = 10g. + u dy - 13dus
+ 13 49
1)dy = Vdy + dy dy + yddys
10y + dy
10 By + 1 β dy + 12 β ddus = 100
posita c constante
10y + 11dy + 12 ady
sit v- - 12
10 39 + 1 βdy = - 12 βddy - dys (d (1β) + 100
du ly = v(Z dys: G3 = var + ldv
ep = + 12dp Ce + Mpleo + 12rd9
10t - 12 d4
4) = - der
β - 11 + d12
- d ( B) dy erit   d(. () 10B = d (n β) = dd (11(3)
sit  si esset 10 = d 11
y = B _ Z. fat +0Z + 5 b
vy 2 u dy
10g c0
+ 12p co. + 129deo + 12t de
+12dgc0
c - (2dC = 12V
12 p + 1rdq - 19, 10t - 12 dr - 12 dp- 109, 12F - 12 dt = 121
deo (12p  (12r r12dt - ut) + 10. 129l- 12. 129 dY)  des (ert (10t - 12dr) + 12r (nr + 12s= 1. t)
F 12r+idt +uL - 1 2p. + 12d9 - 19
10t - 12 dr
t12d 124u toque
1219r + indC et r 1012 ql. n 2 gur
10 a2 tF - 12. 12 t dr + n. 2er + 12. 12rdt + 11. 12 1F
a
co
 unpur ta= p pr ut g 1alus
du = dv b
ddo = dudv b. b sv
dav = audo + ddo
dduo fit v = B N 
toa D
a + 12 X
Bco
12 Agco
12 d p + c2pdv + tadg =0

## Folio 283v — 00068196:0562
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 282r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000562.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 48 lines, status machine
bv + c Co + e du + fdco +g =0
fiet de = 58 1βy + 47Z +βdy + 9 dZ
Ext meo +n dv+ pdco +4=0 Sit v - βg+) Z Con dy2
itaque explicatione facta, ut dus et az simul po sint tolli, debet cum sid una aequatione sit:
eβ dy + C) dZ
aequatione fd dy + fBd2
in altera n(du + n) dZ
ita simul tollentur dy et du
sit ergo sic
p 5 dy + pBd2
bv + cy + e do + fdg + 9 = 0 lv + my +ndv + pdyt q= 0
fiat v= y ( et gC βy + 7 Z, et fiet
debet esse eβt.
bβy + b7Z + eBdus + e) dZ + aag = 6 (3y + lBZ
ca e0
edBy edy Z
ep +fd
L, m, ny p, G)
Ect 
= n B+ p
n7 +pd
s lullmme
vet facere d.
ila manet co
napG + n a Z 1 MBdus P) d
y tapdys
taa
dividatur aequatio prior per  e), et posterior per nγ, debet esse eβtaf-nBtap
seu  enBrasn = en β+ ap  aep. sed ita evanescint  C)
Ergo redeundum ad primas duas arquitiones ut  e +50
+ s
= n3+ pd et fiet
27+pD)
(en BV) sn VS + ep BD (+ spS B = (en B+ A p)  + su pd ( fp sB et fiet
- fnyd fpdE, fnE- epE seu B = sns 
fnv fpE
sp in 
x =  eb - fn, Ergo erit vel ep- sn =0, vel β38 = 7 
e
(3D
2
prins locum habet in casu speciali posterius dabit B ad V, ut d ad  quod non licet
c
ep - fn
Haec ergo via non succedit, nec succedere debet, nam nimium
praestaret
deo = ydd + Zd8 + d dy +  d2
b, c, e, f, 4 sunt dutae si quaeela