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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 12r, page id 00068199:0023
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Habemus solutionom hujus aequationis ilus =  10+ ny fit endrm y = Bsit s dx 10
proponatur jam dy = 10 + 11y + 13 dy fict y = sd ( + 5. 13 ddus
BSI
sudx
BSi
itaque si 13: bs= sit constrns C fet y = bs, (10 ( 14 du 2, Ergo (3 13(b
5F
SbS
Ergo si 13: _ bs sit constans potest solvi aequatio dy = 10 + 1y + (3 ddy
Efficic Tentemus hoc efficere. Sit aequatio data  dy = 10 + 11y + 13 ddy
fiat y = p pV et dy = dpv + pdx et ddy = ddp V + 2dp dv + podr. Et habemus
dv =  1 dp v + pdx = 10 + 1pV + ddp V +, 2 dpdv + p ddV
o
seu de = 1e + _∞ dp = a p +d dp - dp 1 + 10 + P - ddV
- ex
P = 2,  
P- 2dp3 p- 2dp13
+pV cempendio dv = 20 + 2 B + 23 dd v
debet esse 23: bs constans. Faciamus p esse constantem et fiet
23 = 13 et bs2t = bt IPsa, ita nihil lucrati sumus sed si p si ab. - 0
fit 23= 13P :, p - 2. 13dp et 21 = u1B - dp, :, p - 2 13 dp
et quia 23: bt  c constanti fiet d23 = 21 b = 21. 23:c
et
d23: 23 = 21 A: c ergo tantum opus, summari posse 21
Merentietur 23 ipsius 23 valor, habebitur landem aeq. debe
sed debet p esse. 4oxx + 4 1X + 42. ta enim dap = 0
sit 23 = 13, 40XX + 134IX + 13. 42 unde habetur et d23 habetur et 21
+40 Xx + A1X + 4 2
413. 40X - 3. 13. 41
et deberet aeg. fier identica
sed tunc debere determinatur relatio
inter 1 et 13.
aqualio 12g9 + 9de L 1209y + 12 ydy + 102 dydy est aequatio quae solici potest
nam fiat y = bsp, et fit 120 f b5~ bs + 112 bP v b V + 102 vv bN (SV = 0
adeoque 1 120 + 112V + 102 rV = 0 sit _ y=
sit aequatio 100 + AIy + H2 + d 121 4y  112 dy + 22)y = 0
fiat y = 6 Z+c0 et fiet 100ZZ - 2. 1007CD+ 100 coc + 111 (Z  = Coc0) + 1212  Y 212Z COH 12IC0
c0
 (Z () (dZ + d0)  (ZYC) (dq dec) 12 dco - 2cod2)
Ponamus esse 12 =0 fiet dy : y = Y( ( 120: 102) et log y = ( Y (= 120: 102)
(SV.
ra
1099 = N27 1209y + 102 dyddy = 0 fit 120bs bs = 102 v b b v, 12 vdro
y = p SV dy = v (V ddy - vo bs + du Bsv
y = fdx Z s
sit aequatio
102 Vb W ( + 102 pb v 1B ( V = 0
seu 120 + 102 V + 102 V dv
sit aequatio y + 11 dy + 12 ddus + d 121 dydy + 122 dy ddy = 0
fiet du = A2: Z unde habetur aequatio, in qua ipsius Z affectiones ubique
aeque altae. unde faciendo Z- bso habetur aequalio  in qua V et du
10g seu ( = logZ = y et Hinc methodus gentralis omnem
aequationem differentio differentialem in qua abest  unius
literae terminus a affectto ordinaria, atteliendi ad diterentialem
ddy: dd dav: dd7 = dvdv+ d2dZ,: deodeo. sit dv: deo =  dt: t ddv = deo tdct- 1 at
tandemque dcot dat = duo dtat, deodeo = ddZ dtdt + tt dq dZ aZ. facendo t = bs ellemus quidem
e o ui c otanate atis