Pagina:Leonhard Euler Letter 1765-03-31.pdf/5

En Theorema Analyticum continens Criterium, utrur? formula differentialis differentialia ??? gradus ??, sit? integrabilia rec ne! C?????que gradus f???it formula differentialis propo?it??, ea ponendo?

$\delta y=p\delta x;\delta p=q\delta x;\delta q=r\delta x;$: ch. sernper a han? fromam revocatur

$\gamma \delta x$in qua eist? $\gamma$ quantitas ex litteris? x, y, p, q, r ch. ut?ungque con

flata, quoe? ergo mor? sotio differentiata pnebebit? takem formami

$\delta \gamma =M\delta x+\int y\delta y?+P\delta p+Q\delta q+R\delta v$et.

Jam dico quoties formula $\gamma \delta x$ integationem admiltit, totris?

fore? $?-{\frac {\delta P}{\delta x}}+{\frac {\delta \delta Q}{\delta x^{2}}}-{\frac {\delta ^{3}R}{\delta x^{3}}}+ch=0$ a? viuis? seni?.

Exemplum. Sit proposita h??? forumla ${\frac {y\delta x\delta y.x\delta y^{2}+xy\delta \delta y}{yy\delta x}}$ s?mto \delta x consta???

q??e per positiones factas abit in ${\frac {yp-zpp+xyq}{yy}}\delta x$. ut sit $\gamma ={\frac {p}{y}}-{\frac {xpp}{yy}}+{\frac {xq}{y}}$

hac quantitas differentietur, fuet que

$M=-{\frac {pp}{yy}}+{\frac {q}{y}};N={\frac {-p}{yy}}-{\frac {2xpp}{y^{3}}}???;P={\frac {1}{y}}-{\frac {2xp}{yy}};Q={\frac {x}{y}};R=0,S=0??-{\frac {xq}{yy}}$

Hinc sit $\delta P={\frac {\delta y}{yy}}-{\frac {2p\delta x}{yy}}-{\frac {2x\delta p}{yy}}+{\frac {4?xp\delta y}{y^{3}}}={\frac {-p\delta x}{yy}}-{\frac {2xq\delta x}{yy}}-{\frac {2xq\delta x}{yy}}+{\frac {4xpp\delta x}{y^{3}}}$

et ${\frac {\delta P}{\delta x}}={\frac {-3p}{yy}}-{\frac {2xq}{yy}}+{\frac {4xpp}{y**3}}$

Pono $\delta Q={\frac {\delta x}{y}}-{\frac {x\delta y}{yy}}={\frac {\delta x}{y}}-{\frac {xp\delta x}{yy}}unde{\frac {\delta Q}{\delta x}}={\frac {1}{y}}-{\frac {xp}{yy}}$. et ?enus differentiando

${\frac {\delta \delta Q}{\delta x}}={\frac {-\delta y}{yy}}-{\frac {p\delta x}{yy}}-{\frac {x\delta p}{yy}}+{\frac {2xp\delta y}{y^{3}}}={\frac {-p\delta x}{yy}}-{\frac {p\delta x}{yy}}-{\frac {c?\delta x}{yy}}+{\frac {2xpp\delta x}{y^{3}}}$

ideoq ${\frac {\delta \delta Q}{\delta x^{2}}}={\frac {-2p}{yy}}-{\frac {xq}{yy}}+{\frac {zxpp}{y^{3}}}$. Zuare habebit??

$N-{\frac {\delta P}{\delta x}}+{\frac {\delta \delta Q}{\delta x^{2}}}-??={\frac {-p}{yy}}+{\frac {2xpp}{y^{3}}}+{\frac {xq}{yy}}=0$. Ergo formula $\gamma \delta x$ eit integrabilis

$+{\frac {3p}{yy}}-{\frac {4xpp}{y^{3}}}+{\frac {2xq}{yy}}$ Manifesto autem integrale eit

${\frac {-2p}{yy}}+{\frac {2xpp}{y^{3}}}-{\frac {xq}{yy}}$ \frac{x \delta y}{y \delta x}