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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 37r, page id 00068196:0073
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sitit duae aequationes differentiales simplices ad litoras X, y, V, et sit dx
constans, simplices voco ubi ordinatae non du habent affectiones invicem ductas.
Aequationes ita stabunt 1004 1oydx + 101Vdx + 120c dy 120 a dy + 102a dv = 0
u
Et
latens
200aX + 210y dx + 201 vdx - 220 ady + 202 a dv = 0
videndum est an liceat obtinere Equidem non licet efficere, ut simul tollantur do et dus
alioqui etiam evanesceret dx. illud tentandum an liceat simul tollere
Y et de sit  sed vibeo si v facias verb. gr. BZ, perinde esse ac si una litera
quid ergo si faciamus y =  + pv:a, et habebimus
100a +X + 110ZdX +120a dZ + 110sp: a) vdx + 120p dv = 0
102 a de
101.  dX
120 V ap
Et pari jue 20 adx + 210Z dx + 2eo adz + 210 p: a) vax + 220 pdv =0
Lcoradv
201 vdx
Hinc poterunt V et dx simul
tolli si sit
110 p ix +101dX + 120adp 12OP H102a
210pdx + 3o1a dx+ 220 adp 22op, 202a
Huo. Propp dx + 110. 202a pax + 101. 202 aacx (+ 120. 20   10 2
220 D ap
0 = - 120. 210... . . . 210. 102.. . . - 201. 102....-220. 120 a pd p D2.
+101,220....
ita res reducta est ad aequationem disserentialem unique - 207. 120...
Hic apparet aute omnia non licere destruere terminum ep, aut aadx, aut aado
indetermi si indeterminatae
nam prae supponeretur quaesitum, si enim esset 10 ad 120, ut 210 ad 220, vel tale quid utique
utiue simul tolli posset y et dus; idemque est in reliquis; sola ergo aequatio praestanda
superesset 110. 202 - 210. 102 + 101. 220 = 2B1. 120- sed hoc sublato non juvamur nempe termino
apdx, non juvamur. sit aequatio P
sit aequatio, 13 pp ex + 4 apdx + 11 a dp + 10 aa dx =0
data resolveni
Assumatur aequatio e pdx t deadx
sed ita in producto non coanescit pax
sit quid si faciamus y = Z+ p. a et  = qu:a, prodiibis ex duabus datis initiaeqqu.
2t pdx +21 adp +20 adX multiplicanda per 31p + 30a,
p Et fiet
100 a dx + 110 Z dx +120 adz + 11osp: a) Hx + 120( pduvia, = c q- X 120 + β220 p
101(9. a).
120l: a) Dax
102 (a: a) d d9
102 4du
200 adx +210ZAX 1 220 adZ + 210(p. a) Wdx + 220 p du
201 (9: a) Wdx
220(a: a) Wdp
202 1a. ) Wdilq
202 9du
a102 + β. 202 fas
((ano+KY 1OUxL a. 102iF - -
620 + B, p. 201 45. 202a
firt, 110. 220 ppdx + 101. 220pqdx (+ 120. 220 pdp) + 102. 220 pdg
Ergo ut tolli possit sumul 1 et de
iist
+101. 202 99dx F110. 2029dx + 102. 2029dq) + 10. 202 9 dp
 hot a voirapi notpiron ip i
+
Pdlus-qde
dFa-
 paq-gap = sp a x
pags -gupe- F 44
p9
d(9. p)
210. 120 ppdx - 201. 120 poax - 220. 120 pdp) - 202. 720 pdq
-201. 102 99 dx - 210. 102 p9dx - 202, 102 9d97 = 220. 1029dp
Haec aequatio fiet solubilis, si ponamus p habero rationem ad q co
q
imo B et 2 a multi plus
catae
ter
C
omantutur uvicem
pet a suop
ante ut a ad Y
seu esse q = rp: a et soluta res erit per quadraturam quod miror videtur tame spes esse spucessus
si Y esset = a Coanescerent paq et gdp, adeoque et  dx, et p ut 4 rediceentue
omnia ad terminos cognitos. Imo video nec hoc licere ut fiat qLrp: a, natu
ila everescerent hoc loco pdus et qap, quia paus fart = qup inio idem foret esi x non esset onstans
Dfert tamen adhuc se aliquid quod spem facit, aequatio C divellatur in
duas 110. 220 ppolx + 101. 20299 dx) - 210. 120 ppdX) - 201. 10299x) = 30 ac X) c
d(G: p)
seu vy2=
et +1BI. 220 dx = fivc. 220. 10 20V) +102. 220)
101. 220 + 10. 202- 201120 - M. 12
102. 220 - 120. 202
Ergo aequ Patus dividamus per P4 Et sane si aeg.  dividamus per p videbimus revera non adesse
- 120, 2025
sed nec hoc procederet quia ita bis haberetur q: P
duas arbitrarias sed unam p: 9, et habituros nos (p.9) dx, (1: p) ax, dx, aequet L  (q. p
4 u. 202
= 201. 120
2g. 102
nihilus adeo lucrati sumus
aqua tio O reducitur ad 4q
tn almide haec lucta bilia quia p et 9 servant inter se lere q.p
q et d  ande nuilor licet assumere allerius cu
homo glaeom
scolis sit determinata
domultre ie
ulieissimulae eu
in oeffud