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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 247r, page id 00068196:0489
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Notavi jam alibi si aequationem differentialibus
et suminis implicatam eo reducere possim, ut aliqua ex
terminabit alterutra exius determinatis vel affectionibus ejus
in quolibet nembro reperiatur simplex tunc explicari posse
infinitos
redira d senem  adhibita feric perveniri ad
aequationem identicam  cujus quaelibet columna habeat
definitum aunerum membrorum. Unde facilior solito solutio
est, in reductio ad simpliciorem.
Nempe 
erurqu dus = ady + bddys + c d’ y+ e d’y ete =tyf fg +
sty
Aosudus 4 12 513 ddy + 14 sisay et
hsa Cotius +Eo susitu 12 a 13 si4 si r5 CC
d0
lsm sn spu Y 30 (31 B2sdus 13A (35 sb p err
i refert, quomodo literae _a, bc et
=
c
asectio intens
modo membra sint
finita,
tunc bractabilissima
fit series.
etiam determinatis formentum
ogitavi jam, an non proposita aequatio differentialis, licet
et numeri
p
si
adhibitis
alterius naturae huc ad  hanc formam reduci possit
in exemplum
et videtur pes succedere
gy + XX; ay + aY = dy: dx seu yy dx + XX dx = ayaytaxdy
Hanc sum differe summando fit (yydu + LX = ragu + a sxdus.
Haec dividatur per y fit Syyax:y + X: 3y _= ay: 2 + a (xdus: y. quam
4).
differentiando fit vydx: y - dy syydx:y9 + X dx: y - dyX: 3yy = ady : 2 + aX dy. 9 - adipli
C
Tollamus Xdx et X, ut affectiones ipsius X habeantur sola
dxF dus Sygax +yxdx= X dy : 3 =(ayydy: 2) + Ca yxdys e = adysXdus Et prodit aeque
seu multiplicatis onnilbus per yy fit
(ay dy(t axydu (=y x, ( agydy: 2) asndys duys 4 Sygax dy)
a
poraeq.
quod si praeterca differentremus vyipsam 2
ideulica
ubi ora
Oestruuntu
sed ita in govas
Ex  fiet yy adx +2y dydx + 2X dx + x ddx = a ydadus t ady dx incidimms tollendas
illud quidem commode potest obtineri, ut omnibus sublatis potentiis
solae extent in aequatione X, y, dx, dus, ddx, ddy, et C, sed ut non ducantur
in se invicem obtinere nisi per seriem infinitam non licet
sit y: a = CRX 1 X +   + 13 dix + 14 d X + 15 d X etc
Ergus do. dx - 1 dx
i sic y = 10 X +
iet
x
X = aydus - yydx: - ady + X dx
x
+x3
ubi illud memorabire
quod ipsius X progressi
manet indeterminati
sed video rem
non procedere,
quia e possit
explicari dux
pro arttrio
dx = 4y do a dy + Xax, ayddyt acis - gy dax - 2 ydylx- aydys -yylx ddtus 4x dlx + x
a dy - 2aX dy dx + x +x
aydy, actys- 2axdy dx + Xdi = yy adus- Xax, agadys tactus -yy x- Ey ayex ay agaygax, as pt
B sua dus 1cte 3)
adue  adue pax= aydus- yycx osadus- Xdx, ay adyratis agrat. ry titr e
ady, adisraxydxex
2. aydy. yyax, adtus- xdox-dx