第9章 9.14

(1)

$xy'\cos\frac{y}{x} = y\cos\frac{y}{x} + x$

$y'\cos\frac{y}{x} = \cos\frac{y}{x} + 1$

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

$\left(x\frac{du}{dx} + u\right)\cos u = u\cos u + 1 \quad x\frac{du}{dx} = \frac{1}{\cos u}$

$\int \cos u\, du = \int \frac{1}{x}\, dx \quad \sin u = \log x + C \quad \text{元に戻して}$

$\sin\frac{y}{x} = \log x + C \quad （C \text{は任意定数}）$

(2)

$x^2 y' = y^2 - xy$

$\frac{y}{x}y' = \frac{y}{x} - 1 \quad \frac{y}{x} = u \text{ とおくと} \quad \frac{dy}{dx} = u + x\frac{du}{dx} \quad \text{これから}$

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

$\int \frac{1}{2}\left(\frac{1}{u-2} - \frac{1}{u}\right)du = \int \frac{1}{x}\, dx \quad \frac{1}{2}\log\frac{u-2}{u} = \log x + C_1 \quad 2C_1 = C_2 \text{ とおくと}$

$\log\frac{u-2}{u} - \log x^2 = C_2 \quad e^{C_2} = C \text{ とおくと}$

$\frac{u-2}{ux^2} = C \quad \text{元に戻して}$

$\frac{y-2x}{x \cdot yx} = C \quad y - 2x = Cx^2 y \quad （C \text{は任意定数}）$

(3)

$(x^2 - 3y^2)x + 3(x^2 - y^2)yy' = 0$

$(1 - 3\frac{y^2}{x^2})x + (3 - \frac{y^2}{x^2})y\frac{dy}{dx} = 0 \quad (1 - 3\frac{y^2}{x^2}) + (3 - \frac{y^2}{x^2})\frac{y}{x} \cdot \frac{dy}{dx} = 0$

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

$(1 - 3u^2) + (3 - u^2)u(x\frac{du}{dx} + u) = 0 \quad u(3 - u^2)(x\frac{du}{dx} + u) = 3u^2 - 1$

$x\frac{du}{dx} + u = \frac{3u^2 - 1}{3u - u^3} \quad x\frac{du}{dx} = \frac{u^4 - 1}{3u - u^3}$

$\int \frac{3u - u^3}{u^4 - 1}du = \int \frac{1}{x}dx \quad \int \left(\frac{-2u}{u^2 + 1} + \frac{u}{u^2 - 1}\right)du = \int \frac{1}{x}dx$

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

$\frac{u^2 - 1}{x^2(u^2 + 1)^2} = C \quad \text{元に戻して整理すると}$

$C(x^2 - y^2) = (x^2 + y^2)^2 \quad \text{（C \text{は任意定数}）}$