第9章 9.26

(1)

$xy' = 2y - x$ $y' - \frac{2}{x}y = -1$

$P = -\frac{2}{x}$、$Q = -1$ とおくと

$\int Pdx = -2\log x = \log x^{-2}$ これから

$y = x^2 \left\{\int{-\frac{1}{x^2}} dx + C\right\} = x^2 \left(\frac{1}{x} + C\right) = x + Cx^2$ （C は任意定数）

(2)

$2xyy' = y^2 - x^2$ $y' = \frac{1}{2} \cdot \frac{y}{x} - \frac{1}{2} \cdot \frac{x}{y}$

$\frac{y}{x} = u$ とおくと $\frac{dy}{dx} = u + x\frac{du}{dx}$ これから

$\left(u + x\frac{du}{dx}\right) = \frac{1}{2}u - \frac{1}{2u}$ $x\frac{du}{dx} = \frac{u^2 - 1 - 2u^2}{2u} = \frac{-u^2 - 1}{2u}$

$\int \frac{-2u}{u^2+1} du = \int \frac{1}{x}dx$ $-\log(u^2 + 1) = \log x + C_1$ $e^{-C_1} = C$ とおくと

$x(u^2 + 1) = C$ 元に戻して

$x\left(\frac{y^2}{x^2} + 1\right) = C$ $x^2 + y^2 = Cx$ （C は任意定数）

(3)

$y' + \frac{y}{x} = \log x$

$P = \frac{1}{x}$、$Q = \log x$ とおくと $\int Pdx = \log x$ これから

$y = \frac{1}{x}\left\{\int x\log xdx + C\right\} = \frac{1}{x}\left[\frac{1}{2}x^2 \log x - \int \frac{1}{2}xdx + C\right] = \frac{1}{x}\left[\frac{1}{2}x^2 \log x - \frac{1}{4}x^2 + C\right]$

$= \frac{1}{2}x\log x - \frac{1}{4}x + \frac{C}{x}$ （C は任意定数）

(4)

$y'-y \tan x = \sin x$

$P = - \tan x$、$Q = \sin x$ とおくと $\int Pdx = \int - \tan xdx = \int - \frac{\sin x}{\cos x} dx = \log(\cos x)$ これから

$y = \frac{1}{\cos x} \left\{\int \cos x \cdot \sin xdx + C\right\}$ $\cos x = t$ とおくと

$= \frac{1}{\cos x}\left\{\int - tdt + C\right\} = \frac{1}{\cos x}\left\{- \frac{1}{2} t^2 + C\right\}$ 元に戻して

$= - \frac{1}{2} \cos x + \frac{C}{\cos x}$ （$C$ は任意定数）