Novi commentari Academiae Scientiarum Imperialis Petropolitanae . t;^)ydy + {b-ik)y-dy -\- ky^dy -zzo^ quod paradigma ipfum iam ad integrandos , fine pracuia inacterminatorumfeparatione inferuire poteft. Sit ex. gr. hoc modo iii-tegrandum 3J^a, — sydx — 3 dy -j^ ^y dy -\- i sy^dy— sy^dy rr 0. Obferuabitur ex comprtrationc termino-rnm homogeneorura, > fll g -:. 3 , (? _- — <f, f— — a,h ~ o^ b ~ 0^ : :z: o^ e ?::: . ..itegrale erit hoc, x H-j n: log ^l^y • ;cp. 2r^2 DISSERTATIO DE FROBLEMATIBFS^ §. 19. Frobkma V. Integrare aequationcm haneV:adx-\-bdy -H cxdx -\- eydy -^f
Novi commentari Academiae Scientiarum Imperialis Petropolitanae . t;^)ydy + {b-ik)y-dy -\- ky^dy -zzo^ quod paradigma ipfum iam ad integrandos , fine pracuia inacterminatorumfeparatione inferuire poteft. Sit ex. gr. hoc modo iii-tegrandum 3J^a, — sydx — 3 dy -j^ ^y dy -\- i sy^dy— sy^dy rr 0. Obferuabitur ex comprtrationc termino-rnm homogeneorura, > fll g -:. 3 , (? _- — <f, f— — a,h ~ o^ b ~ 0^ : :z: o^ e ?::: . ..itegrale erit hoc, x H-j n: log ^l^y • ;cp. 2r^2 DISSERTATIO DE FROBLEMATIBFS^ §. 19. Frobkma V. Integrare aequationcm haneV:adx-\-bdy -H cxdx -\- eydy -^fydx -{-gxdy — 0. .Fadiim hoc quidem iam elt aliquot modis a lac. Her-manno , iii Commentar. Acad, Jcient. Petrop. Tomo I; fed idem hoc problema breuiori via fequentemin modum folui pofle exiflimo. Pouantur termini pes; ?dx multiplicati .a-\-cx~\-Jyz::u, nec nou termini ?per dy multiplicati b-h ey-\-gx — t :, obtinebituribi y — ? J-.^ atque- hic y — ^^^^^ ; tum aequando:hos duos ifalores, et breuitatis cauflli ponendo,. deriuabitur x r:^ ai - ^ U-\- $; y :=z — y\ t-\-zu- ^^£t aequatio propofita mutabitur in hanc: d t — ^ ud u — -v] t dt -\- £ t d u—: o, cft enim ob «7 -i- cx -\-f y ~ «, et b-^r-ey-^-gx-f^acquatio propofica />n/«(? talis, udx-{-tdy:=zo-, Qife-imdo, dx — oidt — ^du , dy zz ~y\dt -{-. edu, qui-bus diffcrentialibus (iibditutis ia priori- aequatione emer-§it propofua. Haec autem eft homogenea, adeoque ovm^sD. :- ist- artificio M. BernouUi}, vid. Comin:ritar. Acad. Vetrop. tomo I, p. 169, variabiles, hic inter fepermixtiis, tenet fepanibiles; qiiam methodiim mngis adiwc vniucrlalem reddidlt Il/ujlr. Dom. Goldbach, 1. ^ 207. Qiiod cum prirao intuitu non appareret, ab initio ^ fj — ;<•, h-i-ey-^-g^x—t^u^^et deprehendi tum demum , homogeneitatem aequationiproporitiie conciliari pofle, fi fiant mzri^ n^i, hac itaque aequatione homogenea fu
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