Pagina:Leonhard Euler Letter 1765-03-31.pdf/3

pae omnes ita sunt com?arata ut per quam?? functionem multiplicata

reddanta? integrabile?. i?eluti si P et Q farcit functiones homogne??

ipfanem? x et y ejcis? dem dimention r??m numeri. aequatio $P\delta x+Q\delta y$ eo?

integrabilis redditels? divisa p?r $Px+Qy$. Semper en??ni formul??.

${\frac {P\delta x+Q\delta y}{Px+Qy}}$ integrale exhiberi ?j?o?s??t. Sive? algebraice si?? brant*dentem.

Totum ergo negotium integrandi semp?r es sedit? ut pro qua cunque aqua.

tione differintiali proposita talis multiplcator inveniatim quem quand??

functionem sinitam. quandoque vero si differentialia adserit? altioni? qe????

ipsam differentialiainvol?ventem e?se obfe?a?i. Pro aequatione quiderro

$dxxy\delta \delta y+cxy\delta x\delta y+bxx\delta y**y+ayy\delta x**x=0$ f??le jierspicer?? licet dai

multiplicatorem forma ??? $x^{m}y^{n}:$ multiplicatione iqit? fac?a

haberu: $\int x^{m+2}y^{n+1}\delta \delta y+cx^{m+1}y^{n+1}\delta x\delta y+bx^{m+2}y^{n}\delta y^{2}+ax^{m}y^{n+2}\delta x**x=0$

ac integralis quidem primum membriem certo e?it $\int x^{m+2}y^{n+1}\delta y$

sit reliqua pam $\gamma \delta x$ idcoq? integrale completeum $\int x^{m+2}y^{n+1}?????=C\delta y$

fietqo? $\delta \Gamma \delta x=+?cx^{m+1}y^{n+1}\delta x\delta y+bx^{m+2}y^{n}\delta y**2+ax^{m}y^{n+2}\delta x^{2}$

$-(m+2)f-(n+1)f$

ubi nece?s?rio este debet $(n+1)f=b$ ut habcat?? per \delta x ???? dendo

$\delta \gamma =(c-(m+?)f)x^{(}m+1)y^{n+1}\delta y+ax^{m}y^{n+2}\delta x$. unde? $\gamma ={\frac {a}{m+1}}x^{m+1}y^{n+2}$

et ${\frac {(n+2)a}{m+1}}={\frac {c-f?V((c-f))^{2}A(n+2)af)}{zf}}.$ Definitir ergo m et n ce?uatio

unteqialis elit $fx^{m+2}y^{n+1}\delta y{\frac {a}{m+1}}x^{m+1}y^{n+2}\delta x=C\delta x$ ???

${\frac {C\delta x}{x^{m+2}y^{n+2}}}=f{\dot {\frac {\delta y}{y}}}+{\frac {\alpha }{m+1}}{\dot {\frac {\delta x}{x}}}$ i quie? per $x^{\frac {a}{m+1}}y^{f}$ feu? $x^{{\frac {\lambda a}{m+1}}y^{\lambda f}}$ multiplicata

denuo fit integrabilis fi modo ???????? $\lambda f=n+2???\lambda ={\frac {n+2}{f}}$, quia fit

${\frac {(m+1)\delta C}{(n+2)a-(m+?)(???)f}}x^{{\frac {(n+2)a}{(m+1)f}}-m-1}={\frac {f}{n+2}}x^{(n+2)a}{(m+1)f}y^{n+2}+D$ vel? mutatri? constanbilus