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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 8r, page id 00068196:0015
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quaeritur solutio hujus aequationis differentialis
primi gradus dy: de  yy + xx,: aa. compendio scribamus
=dy: de =O: D  e ubi C, et D datae, assumumuus quaesitam aequationem
dy: dx Z 2: T. fiet Ot D4. et disserentiando +C Ax + C at dx- 4 d ) d - D d. E dx
= -  dOdy - C d h dy + 2 = D dy + D 4 dy, seu fiet dy: dx-,Oth. 2 d D. 944:
, +0 + Odh - 2 +D- 9 d 4 = C:D. Ergo landem fiet
Oh+O +OOdt- 02 +9-0) 44 = D540 +0) dG- D4 d D - D7 42.
Si  hoc loco C = yy + XX dO = 2Xdx, dO = 2ydy D = aa d3 =0 4) =0
qu
sit pro quaesitis 4 = 30 3  + n9 ayyt 3 et t sit 41) +4 a, fietque
d4 = y+30 + aysd31 + a) d3 Va 13) a 2 = 3. 307y dy + 2 31) dy = 1. 32
dt =aa egy dgo + ay dq + aa d4u dG =
Ergo liis vatoribus substitutis in aequ. 6. fiet
1. 4 aidys
(+ ad33
(yy + XX) (41ay +42aa) 2X dx + ( + 2yy XX + X) a (ay d9 1 F aa d 42) - aa (yytX X) (1) ho raytgutaslu
Aqu. aa (4ay F graa) 2y 4 , aa (yy +XX) 1. 41 a 4 = at (3. 307  + 2. 31 . Et ordinando fiet
taa Y d41 + yt ad42 + yy41ay2X + yy 42aa 2X + XX 41 ay2X + XX42Aa2X
aa y5d30 = a3y d31 + 2GyXX aayd41 + 2yy a aad42 + xT a ayd4 + xt a aa dqe
= aagyaa  432 - aagy d33 - au Xx aa) d32 - aa XX ad33
aa xx y3 d30 - aa XX ayy d31
= aa 41ay2y = aqe 4244 = Aa XX 1. 41 a
=0
=aa yg1. 41a + a. 231 ay + 3. 1. 32aa
+ a,38yy
Haec aequatio coincidentienda cum aeq. 4 multiplicata per 50y +51 ipsa ag. 4
explicata dat (yy + XX) (quagu f 41 aay +42 a) = aa (30y + 31ayy + 32 aay =+ 338)
inde
50gqa + 5oy’ 41aay + 50y42 a
+50yX X quagy +5OyXX 4l aay + soyxxqa
+ Et yy quaqu + Sigy 41aay + Syy 42a
+5IXX 40ayy F SIXX 41aay + SIXX42a
- 50 44y 305 - 50949 31yy = 50aay 32aay - 50 aa y 3343
.51443013 -514a31 ayy - 514a 3244y -51 a a33a
Si a l Jaret 28 - 41 bC  sepenieposse Cum 440
omissis multiplicationibus possemus fortasse er sex aequationes coincidentiales
praestare, fiere per solas sex quaesitas 30, 31, 32, 37, 41, 42
(et 41 = 30 + b et 42 = 31 +C et  pro Terminis y et yt, et
 Ex y fiet 2a _ 2041 ax + 2d41 a aXX - a2432 = 41  - 30 a
seu 2. 30 a X + 2.  a b X + 2. d3d aa Xx = a+ d32 = (30a) + a b (- 30a) Heebetur d32
X
= 31 aXX + CaXX = a31. Ita habemus vatorem ipsius d33
an
supersunt duae 5e49. per y et Y, ubi occurrunt 32 et 33, una cum d32 et d33.
Tollantur d32 et d33 per valores inventos et differentientur 32 et 33, pervenietur
ad ede tollentur. 3 32 et 33 prorsus, sed descendetur ad dd30 et dd31 ubi
si velimus collere, dd 3C 3, novissima aequatio descendet ad d30 qua alibi
qualia in simplicibus resolvi posse ostendimus