Question:
Write the correct alternative in the following:
If $y=a x^{n+1}+b x^{-n}$, then $x^{2} \frac{d^{2} y}{d x^{2}}=$
A. $n(n-1) y$
B. $n(n+1) y$
C. ny
D. $n^{2} y$
Solution:
Given:
$y=a x^{n+1}+b x^{-n}$
$\frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}}+(-\mathrm{n}) \mathrm{bx}^{-\mathrm{n}-1}$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{n}(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}-1}+(-\mathrm{n})(-\mathrm{n}-1) \mathrm{bx}^{-\mathrm{n}-2}$
$x^{2} \frac{d^{2} y}{d x^{2}}=x^{2}\left\{n(n+1) a x^{n-1}+(-n)(-n-1) b x^{-n-2}\right\}$
$=n(n+1) a x^{n-1+2}+n(n+1) b x^{-n-2+2}$
$=n(n+1)\left[a x^{n+1}+b x^{-n}\right]$
$=n(n+1) y$