SECTIO III. 519

C o r o 11 . 2 .

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680. Proposita aequatione dy z -\-dx —0 seu

ob radices pzr, — j ;

1 -+. v — 1

et

1 erit vel y~-x + a-, vel x+bivel yxz.'- =L ~ J x-\-c , quae collecta praebent:

f +x ^ a+i+cW ^a-^b-'*i=^ c )x yJr (-aV-=^b'\^c)xx

■^(ab-\-ac~ybc)y-\-(bc -'—^ 1

ab\x-abc~ o

quae aequatio etiam ita exhiberi potest :

y+x s —/yy — g xy—b xx -f- Ay -f- B x -f- C — o

vbi constantes A , B , C ita debent esse comparatae,vt aequatio haec resolutionem in tres simplices ad-mittat.

Exemplum 2.

681. Proposita aequatione differentialiydx —a;'V'(dx I 4-dy I )zi:oeius integrale completum inuenire.

Posito jfrrp fit y— xV(pp-\-r)zz o sit ergoy~ux erit uz=zV{pp- f-i) et vnde per

alteram formulam

k'=-l(p-u)+sw^=-l(p-u)-[dp(p+V(pp+i))

at fdpV(pp-y-i)z=:lpV(iy-pp)-\-ll(p-]r'Y{i +pp))

vnde