ao DE IMTEGRATIONE 



Problema 8. 



$&. Propofita aequatione difFerentiali primi gtt- 

 dus : 



{oL-\-$x-\-yy)dx-\-($-\-tx-\-2 ) y)dy — <* 

 inuenire multiphcarores, qui eam reddant integrabileiru 



Solutio. 



Reducatur haec aequatia ad homogeneitatem po- 

 nendo : 



"i zz t -\-f et y zp u -\-g , vt prodeat 



C* -+- P/-r- V£ ~J-P*-r-y«) <// -h(^-4-e/-t-^-i-rf 



-4-£«)</#~o 

 quae pofito tf-t~p/-r-yg'— o et £-t-e/-r-£s=o# 

 vnde quantitates f et g determinantur , vtique fit ho- 

 mqgenea, fcilicet 



{^t-\-yu)dt-\-{zt-\-^u)duzzo ; 



ideoque per multiplicatorem jYt^y^f^J^u integra- 

 bilis redditur. Hinc inuentis litteris / et g aequatio 

 propofita integrabilis euadet , fi diuidatur per 



p(*-/)*-Hy +o(*-/)(j-£)-K0'-£)* > 



feu per 



pxx-\- {y-*rs)x y+^yy-^pf+y gArtg) x 



-(*%g-l-yf+ef)j-i-Pff-i-(y + e)f& + tgg 

 Cum autem fit fzp^t' et ^ = &$?. 

 prodibit diuifor quaefitus : 



pjc*-f-(V H-e)^-^- 4,^-1 ^yT^P? 



In- 



