第10章 10.2

(1)

$y' = e^x - x$

$y' = P$ とおくと $P' = e^x - x$ $\frac{dP}{dx} = e^x - x$

$\int dP = \int (e^x - x)dx$ $P = e^x - \frac{1}{2}x^2 + C_1$ 元に戻して

$\frac{dy}{dx} = e^x - \frac{1}{2}x^2 + C_1$

$\int dy = \int (e^x - \frac{1}{2}x^2 + C_1)dx$ $y = e^x - \frac{1}{6}x^3 + C_1 x + C_2$ $C_1 = A, C_2 = B$ とおくと

$y = e^x - \frac{1}{6}x^3 + Ax + B$ （A,B は任意定数）

(2)

$y'' = x\cos x$

$y' = P$ とおくと $P' = x\cos x$ $\frac{dP}{dx} = x\cos x$

$\int dP = \int x\cos x dx$ $P = x\sin x - \int \sin x dx = x\sin x + \cos x + C_1$ 元に戻して

$\frac{dy}{dx} = x\sin x + \cos x + C_1$

$\int dy = \int (x\sin x + \cos x + C_1)dx$

$y = -x\cos x + \int \cos x dx + \sin x + C_1 x + C_2 = -x\cos x + \sin x + \sin x + C_1 x + C_2$

$= 2\sin x - x\cos x + C_1 x + C_2$ $C_1 = A, C_2 = B$ とおくと

$y = 2\sin x - x\cos x + Ax + B$ （A,B は任意定数）

(3)

$y''=1-x\log x$

$y'=P$ とおくと $xP'=1-x\log x$ $\frac{dP}{dx}=\frac{1}{x}-\log x$

$\int dP = \int \left(\frac{1}{x}-\log x\right)dx P=\log x - x\log x + x + C_1 \text{元に戻して}$

$\frac{dP}{dx}=\log x - x\log x + x + C_1$

$\int dy = \int (\log x - x\log x + x + C_1)dx$

$y = x\log x - x - \frac{1}{2}x^2\log x + \frac{1}{2}\int xdx + C_1 x + C_2 (C_1-1)=A, C_2=B \text{ とおくと}$

$y = x\log x - \frac{x^2}{2}\log x + \frac{3}{4}x^2 + Ax + B \text{(A, B \text{は任意定数})}$

(4)

$y''=x^2 e^{2x}$

$y'=P$ とおくと $P'=x^2 e^{2x}$ $\frac{dP}{dx}=x^2 e^{2x}$

$\int dP = \int x^2 e^{2x} dx = \frac{1}{2}x^2 e^{2x} - \int xe^{2x} dx = \frac{1}{2}x^2 e^{2x} - \frac{1}{2}xe^{2x} + \int \frac{1}{2}e^{2x}dx$

$= \frac{1}{2}x^2 e^{2x} - \frac{1}{2}xe^{2x} + \frac{1}{4}e^{2x} + C_1 \text{元に戻して}$

$\frac{dy}{dx}=\frac{1}{2}x^2 e^{2x} - \frac{1}{2}xe^{2x} + \frac{1}{4}e^{2x} + C_1$

$\int dy = \int \left(\frac{1}{2}x^2 e^{2x} - \frac{1}{2}xe^{2x} + \frac{1}{4}e^{2x} + C_1\right)dx$

$y = \frac{1}{2}\left[\frac{1}{2}x^2 e^{2x} - \frac{1}{2}xe^{2x} + \frac{1}{4}e^{2x}\right] - \frac{1}{2}\left[\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}\right] + \frac{1}{8}e^{2x} + C_1 x + C_2$

$C_1 = A, C_2 = B$ とおくと

$= \frac{1}{8}[2x^2 - 4x + 3]e^{2x} + Ax + B \text{(A, B \text{は任意定数})}$