第3章 3.13

(1)

$y = xe^x$

$y' = e^x + xe^x$

$y'' = e^x + e^x + xe^x = 2e^x + xe^x$

これから

$y^{(n)} = ne^x + xe^x$

(2)

$y = x^2 e^x$

$y' = 2xe^x + x^2 e^x$

$y'' = 2e^x + 2xe^x + 2xe^x + x^2 e^x = 2e^x + 4xe^x + x^2 e^x$

$y''' = 2e^x + 4e^x + 4xe^x + 2xe^x + x^2 e^x = 6e^x + 6xe^x + x^2 e^x$

$y^{(4)} = 6e^x + 6e^x + 6xe^x + 2xe^x + x^2 e^x = 12e^x + 8xe^x + x^2 e^x$

$y^{(n)} = n(n-1)e^x + 2nxe^x + x^2 e^x$

(3)

$y = e^x \sin x$

$y' = e^x \sin x + e^x \cos x$

$y'' = e^x \sin x + e^x \cos x + e^x \cos x - e^x \sin x = 2e^x \cos x$

$y''' = 2e^x \cos x - 2e^x \sin x$

$y^{(4)} = 2e^x \cos x - 2e^x \sin x - 2e^x \sin x - 2e^x \cos x = -4e^x \sin x$

これから $y^{(n)} = (\sqrt{2})^n e^x \sin\left(x + \frac{\pi}{4} n\right)$

(4)

$y = x \sin x$

$y' = \sin x + x \cos x$

$y'' = \cos x + \cos x - x \sin x = 2\cos x - x \sin x$

これから

$y^{(n)} = n \sin\left\{x + \frac{\pi}{2}(n-1)\right\} + x \sin\left(x + \frac{\pi}{2}n\right)$