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 $0$

$y=a\cos \left(\log x\right)-b\sin \left(\log x\right)$

On differentiating w.r.t $x$, we get

$\therefore \frac{dy}{dx}=\frac{a(-\sin \left(\log x\right))}{x}-\frac{b\left(\cos \left(\log x\right)\right)}{x}$

$=-\frac{\left(a\sin \left(\log x\right)+b\cos \left(\log x\right)\right)}{x}$

$\Rightarrow x\frac{dy}{dx}=-[a\sin \left(\log x\right)+b\cos \left(\log x\right)]$

Again, differentiating w.r.t $x,$ we get

$x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-[\frac{a\cos \left(\log x\right)}{x}-\frac{b\sin \left(\log x\right)}{x}]=-\frac{y}{x}$

$\Rightarrow x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$