Question

$y=\sin \left({\log }_{e}x\right)$ then $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=$

• A. $\sin \left({\log }_{e}x\right)$
• B. $\cos \left({\log }_{e}x\right)$
• C. $y^2$
• D. $-y$

Answer 1

Answer: (D)

$-y$

$y^{{}^{\prime }}=cos\left(lo{g}_{e}x\right)\frac{1}{x}$
$y^{\prime \prime }=\frac{-cos\left(lo{g}_{e}x\right)}{x^2}-\frac{sin\left(lo{g}_{e}x\right)}{x^2}$
$x^2{y}^{\prime \prime }=-x{y}^{{}^{\prime }}-y$

$x^2{y}^{\prime \prime }+x{y}^{{}^{\prime }}=-y$