${\displaystyle\eta(x_{1})=\eta(x_{2})}$
${\displaystyle\overline{y(x)}-y(x)=D(x)}$
${\displaystyle D(x)=\epsilon\cdot \frac{D(x)}{\epsilon}}$
${\displaystyle\frac{D(x)}{\epsilon}=\eta(x)\Rightarrow D(x)=\epsilon\eta(x)}$
${\displaystyle\overline{y(x)}=y(x)+D(x)=y(x)+\epsilon\eta(x)}$
${\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}(\overline y(x))=\frac{\mathrm{d}}{\mathrm{d}x}\left(y(x)+\epsilon\eta(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon \frac{\mathrm{d}}{\mathrm{d}x} \eta(x)}$
${\displaystyle I(\epsilon)=\int_{x_{1}}^{x_{2}}F(x,y(x)+\epsilon\eta(x),\epsilon \frac{\mathrm{d}}{\mathrm{d}x}\eta(x)+\frac{\mathrm{d}}{\mathrm{d}x}y(x))\mathrm{d}x}$
${\displaystyle \frac{\mathrm{d}I(\epsilon)}{\mathrm{d}\epsilon}=\int_{x_{1}}^{x_{2}}\frac{\partial F}{\partial \epsilon}\mathrm{d}x}=0$
${\displaystyle \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial x}\times \cfrac{\partial x}{\partial \epsilon}+\cfrac{\partial F}{\partial(y(x)+\epsilon\eta(x))}\times\cfrac{\partial(y(x)+\epsilon\eta(x))}{\partial \epsilon}+\cfrac{\partial F}{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}\times \cfrac{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}{\partial \epsilon}}$
${\displaystyle \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial(y(x)+\epsilon\eta(x))}\times \eta(x)+\cfrac{\partial F}{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}\times \frac{\mathrm{d}}{\mathrm{d}x}\eta(x)}$
${\displaystyle \epsilon\to 0\Rightarrow \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial(y(x)}\times \eta(x)+\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\times\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)}$
$ \begin{array}{l} {\displaystyle\frac{\mathrm{d}I(\epsilon)}{\mathrm{d}\epsilon}=\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)+\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\mathrm{d}\eta(x)}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\left[\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\eta(x)\right]_{x_{1}}^{x_{2}}-\int_{x_{1}}^{x_{2}}\eta(x)\frac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\left(\cfrac{\partial F}{\partial(y(x))}-\frac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\right)\eta(x)\mathrm{d}x}\\ {\displaystyle =0}\\ {\Rightarrow}\\ \cfrac{\partial F}{\partial(y(x))}-\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}=0\\ \end{array} $
$\cfrac{\mathrm{d}}{\mathrm{d}x}(F(x,y,\cfrac{dy}{dx}))=\cfrac{\partial F}{\partial x}\cfrac{\mathrm{d}x}{\mathrm{d}x}+\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\cfrac{\mathrm{d}(\cfrac{\mathrm{d}y}{\mathrm{d}x})}{\mathrm{d}x}=\cfrac{\partial F}{\partial x}+\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\cfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\cfrac{\mathrm{d}F}{\mathrm{d}x}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}\left(\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial \frac{\mathrm{d}y}{\mathrm{d}x}}\right)=\cfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}+\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}[y]=\cfrac{d\mathrm{F}}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}-\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}$
$=\cfrac{\mathrm{d}F}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}-\cfrac{\mathrm{d}y}{\mathrm{d}x}\left(\cfrac{\partial F}{\partial y}-\cfrac{\partial F}{\partial (\cfrac{\mathrm{d}}{\mathrm{d}x}y(x))}\right)=\cfrac{\mathrm{d}F}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}\left(F-\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\right)=0$
$F-\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}=C$
${\displaystyle\overline{y(x)}-y(x)=D(x)}$
${\displaystyle D(x)=\epsilon\cdot \frac{D(x)}{\epsilon}}$
${\displaystyle\frac{D(x)}{\epsilon}=\eta(x)\Rightarrow D(x)=\epsilon\eta(x)}$
${\displaystyle\overline{y(x)}=y(x)+D(x)=y(x)+\epsilon\eta(x)}$
${\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}(\overline y(x))=\frac{\mathrm{d}}{\mathrm{d}x}\left(y(x)+\epsilon\eta(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon \frac{\mathrm{d}}{\mathrm{d}x} \eta(x)}$
${\displaystyle I(\epsilon)=\int_{x_{1}}^{x_{2}}F(x,y(x)+\epsilon\eta(x),\epsilon \frac{\mathrm{d}}{\mathrm{d}x}\eta(x)+\frac{\mathrm{d}}{\mathrm{d}x}y(x))\mathrm{d}x}$
${\displaystyle \frac{\mathrm{d}I(\epsilon)}{\mathrm{d}\epsilon}=\int_{x_{1}}^{x_{2}}\frac{\partial F}{\partial \epsilon}\mathrm{d}x}=0$
${\displaystyle \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial x}\times \cfrac{\partial x}{\partial \epsilon}+\cfrac{\partial F}{\partial(y(x)+\epsilon\eta(x))}\times\cfrac{\partial(y(x)+\epsilon\eta(x))}{\partial \epsilon}+\cfrac{\partial F}{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}\times \cfrac{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}{\partial \epsilon}}$
${\displaystyle \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial(y(x)+\epsilon\eta(x))}\times \eta(x)+\cfrac{\partial F}{\partial\left(\frac{\mathrm{d}}{\mathrm{d}x}y(x)+\epsilon\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\right)}\times \frac{\mathrm{d}}{\mathrm{d}x}\eta(x)}$
${\displaystyle \epsilon\to 0\Rightarrow \frac{\partial F}{\partial \epsilon}=\cfrac{\partial F}{\partial(y(x)}\times \eta(x)+\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\times\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)}$
$ \begin{array}{l} {\displaystyle\frac{\mathrm{d}I(\epsilon)}{\mathrm{d}\epsilon}=\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)+\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\frac{\mathrm{d}}{\mathrm{d}x}\eta(x)\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\mathrm{d}\eta(x)}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\cfrac{\partial F}{\partial(y(x)}\eta(x)\mathrm{d}x+\left[\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\eta(x)\right]_{x_{1}}^{x_{2}}-\int_{x_{1}}^{x_{2}}\eta(x)\frac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\mathrm{d}x}\\ {\displaystyle =\int_{x_{1}}^{x_{2}}\left(\cfrac{\partial F}{\partial(y(x))}-\frac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}\right)\eta(x)\mathrm{d}x}\\ {\displaystyle =0}\\ {\Rightarrow}\\ \cfrac{\partial F}{\partial(y(x))}-\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\frac{\mathrm{d}}{\mathrm{d}x}y(x))}=0\\ \end{array} $
$\cfrac{\mathrm{d}}{\mathrm{d}x}(F(x,y,\cfrac{dy}{dx}))=\cfrac{\partial F}{\partial x}\cfrac{\mathrm{d}x}{\mathrm{d}x}+\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\cfrac{\mathrm{d}(\cfrac{\mathrm{d}y}{\mathrm{d}x})}{\mathrm{d}x}=\cfrac{\partial F}{\partial x}+\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\cfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\cfrac{\mathrm{d}F}{\mathrm{d}x}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}\left(\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial \frac{\mathrm{d}y}{\mathrm{d}x}}\right)=\cfrac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}+\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}[y]=\cfrac{d\mathrm{F}}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}-\cfrac{\partial F}{\partial y}\cfrac{\mathrm{d}y}{\mathrm{d}x}+\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\mathrm{d}}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}$
$=\cfrac{\mathrm{d}F}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}-\cfrac{\mathrm{d}y}{\mathrm{d}x}\left(\cfrac{\partial F}{\partial y}-\cfrac{\partial F}{\partial (\cfrac{\mathrm{d}}{\mathrm{d}x}y(x))}\right)=\cfrac{\mathrm{d}F}{\mathrm{d}x}-\cfrac{\partial F}{\partial x}$
$\cfrac{\mathrm{d}}{\mathrm{d}x}\left(F-\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}\right)=0$
$F-\cfrac{\mathrm{d}y}{\mathrm{d}x}\cfrac{\partial F}{\partial(\cfrac{\mathrm{d}y}{\mathrm{d}x})}=C$