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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 84v, page id 00068196:0166
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primi
Adesto diferentialis sellendi gradus
dy. sx = aax +  g,: a
dy: dx = A4  yy,: aa seu  aadu = axd + gydx und tandem
aX = aa seu ax  = dy:  - gy  fiet adx = BB. x aby - duydax): dexdx - 2ydy a
Sit   dx dx = a ddx fiet a Hadis = Hacdx) dly- dyddx adtu rydys: aa
sit ddx =0 fiet adx adx dx =aaddus - 2y dxdy, fiet: ex  dxx t g
dpdx + 2(y. a) _ dy F (yy: aa) 1y dy = Gy: aa) dydy + Ally
et p
et dx = Y((y y. aa) dydy + addys) - (y: a) dy Hic res eo redit, ut summari
possit ircationalis, sed ea non habet legem homigeneorum summabilem.
Certum interia est, possibilem esse summationem, si cognosceremus quod
in re est, nempe ralorem ipsius adius: dydus per y:a,  qui determinatus
est in praesenti problemate,  ex posito B ddx- sit ergo addus = gia dydus
edileme  stns Amptricem
relationis genus, quam non nisi una quantitas ingreditur et fret
dx = du + (79 aa +y a) et dx. dy = ax +97, aa = F8 + (yg. aa + 9. a V)
ddx = 0 = ddy H(... . 7 + dy d H(..
dx dx: dyily: = gy. aa + y: ale
et
si (d Cy a y. a, sit Z.a fut de- HY ayay
G.al uydus a
d (y.: a ) -17 Ladx 2dly aus (. eplus - rydus aa
et et _ d y. a 2e = -24 y axta - ryay
= l g: ale= a 2 day (gy + y e,) + 2y dy
d
fiat = d Cy: a) e = 2 a dy y: a (7y + 7 e7)
a
dydys
Arg. 4.9. a2
jam dptx: aydus tellatur, fit
et tollendo adus = adydus y. are
+29tqu, sit y. a e = Z. a
fret 0- 1Z= 2a dy, Z (17 + 2) + rytus sed ita redimus ad d. ffeientialem pumi
gratuis data difficiliorem.
ab initio
dx Utilis tamen e erit pagine prime Methodus, si detur aequatio
differentialis posterioris gradis primatia, in quo
erga Ex his video frustra desen ad aequationem differentiatem 2di
i
gradus ut nostra methodus valeat
sit aequ: dus. dx = XNEgG, ica sit sit  p. a = Z. y Y. a = v: x et fiet.
dy: dx  ax du axtx, XX 2+ de caa XX : ad + aavo :xx
aadus = xxdx + gyx quaeritur ut _ a et affectiones ex  et y
y cum sins affectionibus sivet legeri honogeneorum cum a sit  gra- a, et
X. a = Z: a fint A9. a ve do a et ex a sed ut dixi
aa  = XX + gy, exidy fit o = Exdx + vydus, dxadys +xxty, lys d- dxdtus: dydus
2X dxdys + 2ydydy + xxdys dax + gydus dex - pxdxddus - gy dxddus
seu 2 axdus + ryhe + x  u e  +. dy dix  dus x- ax ddu
XX
XA yY : XX
du ddx e - dxddx - dxddus + dyddus
XX aa = Xx+gy, dx: dy fit 0 = 2Xdx + rydys, dp (idy)
FXx+gy, dys ddx- axdaus, : dyflus)
et ) fiet nydu 2ydx + gyslu ddx- dxddus, : dy dys) 1. 2Xax dx: dy + XX (dus ddx - dxddus, : dydus) = 0
fiat
dividamus per 7F : dy.
un
remanebit dydax-dxaduss dydus, quod si faciamus =0,
dux:  = ddy :ty
do fier a
=9AE
 BE
et logdx = logdy + log. deo posita deo constante
g. 2 t de tid et su - ev
a  unde Z: V = qu s. Z + qu Sd2
fiet Qxrc ay.  = L. X V
4: a = ft ) fdvco X: a = dZ. Ax dv: d2 g: a 2 sare (Z detz: d2a2
X: a = SedZ R g: a = sddque. (V xt dg :dx = dx: dv
fiet sit dy: dx =, XX + 9y, :aa s. rl dx: do = qu. log. AX + qu. Sdv: v, et fiet
dx. d ((2 - qu. los 4X) = (daz. dv et a (+ ( 2 - qu sat2 = a du, et fiet
tindem dique e
su ergo fit d 1a2. do =
=2 d1Z. 12 3 :, 2 Y( 4 = 4 saq2) = ddZ ep
Sactzidz
dex -  
d Xe -exe 1 Ax: 2
A C- 2, fiat  A - da, X.2