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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 151r, page id 00068196:0299
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homveoncolarm
um differentralium rationalium ordinarae
sit aequalio rationalis Generalis deteantia
O= 10 + 11X + INXy + 123, + 12I X9 + 13X, + 1A2XyF 131Xy + 14 X + 12Xy , 41X7 +15X et
1G
1
ee
14B
iffiulset Hujus aequatio diterentialis
CMax + 11Xdy L M XAx 12 12IXydx + D2IX ay +rge ix +, 122 Xyty +3. 131 Xyex + 13. Xay  2 X ax 4. 13CX yaxr2 172  du 13. y   l 1
du x ixgn ite
fye rydy yex yay
xydys yax y dy
gix ydq
0 = 20 dx F 3INay + 21 adr + Z1 Xy Ax + 3_ Xdy + 2 _ X. lx 1 321 Pydy. + 221X y ex + 9X Xay + 23 X sx ~e xisa 13 3. KB) F23, E1B St  4 
dy
conferatur haec aequatio cum data,
y
O2
y dx
Gdy jax xisay yex yay  irix Ryius sex s
1= 20 1 = 31 B = 21: 2 121 = 2112 (9= 32 13= 12. 3 1 - 3112 131= 211. 3 = 33 41 - 23.4 11 (  2 132 C 12 3 . N. 1
dy
yily
yax
141 = 231: 4 = 34 15. 24.5
it gencaliter fit aequatio X
m na Ergo
Et genemaliter sit terminus X Y ejus differentialis M. X =  + n. X
n
n
terminus  in data aequatione, ut sit summabilis debet coefficiens ipsius X divisus per M
nKXd.
aequari coeffirienti A ipsius X y diviso per n, seu n. h X y =n K X. y
Sed ene jam aliam methodum qua pergatur ad transcendentes:
o= 10 + 1X + SR SP s2oRAL+N ) + s2B+ g+  0
30+ 31y
 30+ 3IX
defferentrando +11. 304 + 11. 30. 31 X dx + 11. 30. 31 ydy +,, 31 Xy ex
2. st dy
Y 20. 30
y dx
y1x
+ 21. 30 +20.31 V
Imo videtur 10 et et plane posse omitti
dy
m-1 n
+ 21. 31 ~
C2
Sed  forte
sed quia abest. yu et xX posset forte tale quid adjic
Ae 
30 + 3D 31 X + 31 f habet differentiam 4030. dx +F3 Xdu
Si homoeplosin negligamus fiet sic:
ly
ax
d, 30r71X, gV,  k 
aque nevi gt hrex Tx ydu
10 + HX +ny + (30+ 3IXdx + 540+y Ly + s ndX + 81 dy
Asi
(30 1
60+61y
Chor3X +)2Xy 132
item sic = 10 + UX + inxy + 12X + f(   Ct
50+SIX
unde diferentiaado fiet 1 1. 40 dx + 1. 41 XAX + 1. 41. Xdy +.
9dx
ydu
+ + ing0
F12.40
dgouus
1. 41 X dy +11. 41 X ydx 482
ydx
712. 41
49251) 432 1)
du
40
+ 12 1n
+ 326)
homoeo plota
F 30 5) +31.41) +31 151)
Ergo vodetur quaevis aequatio diffeentialis simplex primio tansentialis simplex se in
qua indeterminata non excedit primum gradum reduci posse ad quadhaturas.
Idem de non homoeoptota derum puto. et subo summaelorio vinento 3.33 Yy +5ds
pro altiori padum ascribemus: e0 121XG + 1353 unde differentiando ad
ad cg. 2. adjicietur.
2. 12. Xydx t 1ux dus ZisX ex
y Ax 3.13) ay
ponamus Si ia servit indicandae
121. 407 2. B21. to rip 3. 17 40 xax
d
2. 11. H X ydx + 2.121XAx + 121. 41 Zdy
Xy dy Xy dys
+ 121. H.
legi homogenioru nec tamen arb, +3.13. 41
a
+3. 335)
y de
y’ay
13. 25(51) +33. 6)
+3. 13. 41Xdx
y dy
+ 33(51)
6 = , 11. 40 x 41. H1 nhy L M HI XdX FuI, Y X L u. 11 Xdy +712. H1 X Ex +3. 12I. H. X ydy + 212141 Xy ax + D21 41 Nay + 3. 13. 4, X dx
33.51
aciter Colligendo in num
+30. 41
11140 2. 12. 570 2.12.41
92. 5i
31. 51
+3.1341
31. S1 3181 2151 121. 40
2iu. 4 84
3. 1940
+3. 33. 51
A 81 = 82 + 11. 40 - 12. 40 B3 = 2. 54 f12. 41 : 111. 41 5b =3.88
13. 33.81
in
ex
idu
E
33.51