y=f(x)y = f_{(x)}y=f(x) とおくと y′=f(x)′y' = f'_{(x)}y′=f(x)′ 点 P(x,y)P(x, y)P(x,y) と点 (0,2y)(0, 2y)(0,2y) を通るので傾き f(x)′f'_{(x)}f(x)′ は f(x)′=−yxf'_{(x)} = -\frac{y}{x}f(x)′=−xy となる dydx=−yx\frac{dy}{dx} = -\frac{y}{x}dxdy=−xy ∫1ydy=−∫1xdxlogy=−(logx+C1)logxy=−C1e−C1=C とおくと\int \frac{1}{y} dy = -\int \frac{1}{x} dx \qquad \log y = -(\log x + C_1) \qquad \log xy = -C_1 \qquad e^{-C_1} = C \ \text{とおくと}∫y1dy=−∫x1dxlogy=−(logx+C1)logxy=−C1e−C1=C とおくと xy=C(Cは任意定数)xy = C \qquad (C \text{は任意定数})xy=C(Cは任意定数)