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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 19r, page id 00068196:0037
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sed hanc autem fermam reduci potest omnit aequatio  in qua i non
e
pliter sit
Au
assagit ultra  ydu et yydx nam semper effici stotes cu aquatione dfferen
tadi, ut tollaram termnus ab ultimus seu obsolatus a quo abest y et du cum
oporteat tantum ponere x  hq et panere unde si adesset ddx.
 Aliter sic: Cly sdy: y +H fdgru + M my + n = 0. fiet
ponere y  v+ tr undem et
Idy + 1fdy: y dy + gy say: y dl F an mdus +yam + dn
fit -5dy: y = l dy + mdy + y dm + dn, : l dy + y dl = my +n,: ly
et liet H U ydy ( Im ydy. + lyyam = lyan = (Im y dyd + In dy + ny dl
lyy +am y +an = 0 firt 2lydyt yydl tamdu Eyam tad = 0
ubi pro gu in aeque 2. substituerdo valorem ex aeq. 1 fiat Ellydus Almdy amy al B aand 20 sit aequalio
 sit aequatio assumta
+ al yam + aal du
cata Aynus +laβdy + _) y ax + ab dx =0 quae multiplicata per MMl: a, ficut
rlLy dy = 22β dus + (Il: a_ Vy dx +, lB dx = 0 firt per dy, am = Elβ et = 2B( al + al am = 2 (V. a) dx
ser - fa3 dl + fl d B3 + fBdl = se GV. 1X et habetur t nam db: t= J dx ad(37 i
in emceia, ut comparatione, jum aequatione
et sit a dB = ) dx quod non est in potestate
perentiatis cum differentiata, multiplicationem differentialis, imo et differentiatae
trans formationem differentialis H malationem figca differentiatae per explicationem
sed ita destruitur ln hac plagula
(2
Conjunxamus zc varia
differentiatam et differentiandam
Aliter lyga amus + a n = b disterentiando 2lydys+ amdus + yy dl + ayam + a du=
e 1
addeitur prima mulsiplicata adiatur securos fitaalydus tam dus rjydl +  ydm + a dn ∞E= 0
conjungendae quationes
sit a aataadl: l = hax et destinetur ju
+sujy ex atimydu +aa hndx
sit et fiet 24lydus+ damdys + Aayam +a dn
et ex aequ. datu ay dys  (e dy + J. ydx + at dx--
cetimyd caex
arius al-ah d
quae mulliplicata per 2l dabil 2alydu = 2 al Bdus + 2l)i ydx +2a Dl ax
et fiet m = am = 2 lB av 21 d 3+ 2B al - 2βd1 = 2 1) 4X
(dy Ah+ hdy
y
=0
yy
ldytyali + an =
et prok sdy substituendo valorem = ly+m - n filo - = 17 =ma - n dh + hh dy et ∞ +
 y
elgrtiy alosidu
seu -lgydh- mydh - ngus dh + hhydy = alu da- amdus thlus dus + h gy dl + hyam + h yy dn =0
ubi si faciamque m=
-
seu +hh y dy - hm dy 
ut hl
= lyy dh - mydh =c
nyu de - hly at
ceton
=hydm
+ h yy
fint haydy
thyy dus
ly dh = hldh = 0 sit jam aequatio data h ydys + h β du + kJy dx + hD dx =
+h yy du
ascribi deberet. gnu...
fit jamt fiat jam in = o, et prodibit
d
β ses et ldh - ndh + hdh + h an = hY da et ld
satiahat
seu  βdh(+ hat -nah + hd (= hah + hdn = h) dx in hac novisima
et fiet h + l
solutio
pettur
si preter n fuisset n+ fy
aequationis daba nam ficiet Sbax = C3h mihh, unde deinde facile habebitur in
dE
liabe
Ergo habetur solutio hujus equationis ydus +bdus + Vyax+ Bd
lx: β3. - h 
est
h y dy + hh dy= ny ahp
hI
aequalis nihilo
st 
+. h G dri
a quatio ne substitinuur ipsius db valor fie
Dda +Bdv+ V B x + vav + n dv +e vdx +H dx = c fiat v = y+ _K
Hinc si B sit constans ve
per aequatione, jam solutas
firt  au= dy+ dK et prodit y dy + kdy + ydK + k dk
X
pa
sit aequatio data
Eyax  Ed
 tEs ) ax
+ adx
= 0 sit jam k+ n = b, seu k = b-
unde n = nesx WV S)dxio- bluta b sich itto posit ti st logur
seu n logii (i dxilogu b log teate posito log W = _) atri h
copto 10 = si et fit n= hphs pax:h + b:
et denique =  bh seu 1= sbb - (Dax et fiit hoc
hsdu + H(= Ddx: dh) + h sdX: h + b: h + y. f = 0 sed
rectum
7, quod sit Sex +l+n + y, cum l+n componant unum
e sumamus erus. hunc
augeamus addendo  p. p y say: y
in soco
et B - - b d n + Eb dx = = n dx L HdX
et (Bax = Bh - e . hh et = bdh - Ma A +
y dy + b dy  - y dn  t-bden = 0 fit Y = E-
utique ex data aeqvatio ne fiet
+8 ydx
nvax
Cabcne
unde absunti
nM: dx = () a  H. n)  Lb
log.
6 f sob 
miu
eodo
y) Ay-m ha
et h=