Question

If $y=a\cos \left(\log x\right)-b\sin \left(\log x\right)$, then the value of $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

Given, $y=\cos \left(\log x\right)-b\sin \left(\log x\right)$
On differentiating w.r.t. $x$, we get
$\frac{dy}{dx}=a\frac{[-\sin (\log x\left)\right]}{x}-\frac{b\cos \left(\log x\right)}{x}$
$=-\frac{\left[a\sin \right(\log x\left)\right]+b\cos \left(\log x\right)}{x}$
$\Rightarrow x\frac{dy}{dx}=-\left[a\sin \right(\log x)+b\cos (\log x\left)\right]$
Again, on differentiating w.r.t. $x$, we get
$x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-[\frac{a\cos \left(\log x\right)}{x}-\frac{b\sin \left(\log x\right)}{x}]=-\frac{y}{x}$
$\Rightarrow x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$