第7章 7.15

(1)

$xy(y-x)=2$

$F=xy^2-x^2y-2$ とおくと

$F_x=y^2-2xy \qquad F_y=2xy-x^2$

$\frac{dy}{dx}=-\frac{y^2-2xy}{2xy-x^2}$

(2)

$e^y-e^x+xy=0$

$F=e^y-e^x+xy$ とおくと

$F_x=-e^x+y \qquad F_y=e^y+x$

$\frac{dy}{dx}=\frac{e^x-y}{e^y+x}$

(3)

$x^y=y^x$

両辺対数を取って

$y\log x=x\log y$

$F=x\log y-y\log x$ とおくと

$F_x=\log y-\frac{y}{x} \qquad F_y=\frac{x}{y}-\log x$

$\frac{dy}{dx}=-\dfrac{\log y-\dfrac{y}{x}}{\dfrac{x}{y}-\log x}$

(4)

$\log\sqrt{x^{2}+y^{2}} = \tan^{-1}\frac{y}{x}$

$F = \log\sqrt{x^{2}+y^{2}} - \tan^{-1}\dfrac{y}{x}$ とおくと

$F_{x} = \dfrac{\dfrac{x}{\sqrt{x^{2}+y^{2}}}}{\sqrt{x^{2}+y^{2}}} - \dfrac{-y}{x^{2}+y^{2}} \qquad\qquad F_{y} = \dfrac{\dfrac{y}{\sqrt{x^{2}+y^{2}}}}{\sqrt{x^{2}+y^{2}}} - \dfrac{x}{x^{2}+y^{2}}$

$\frac{dy}{dx} = -\dfrac{\dfrac{x+y}{x^{2}+y^{2}}}{\dfrac{y-x}{x^{2}+y^{2}}} = \frac{x+y}{x-y}$