Question

If $y=a\cos \left(\log x\right)+b\sin \left(\log x\right)$, then

• A. $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$
• B. $x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-y=0$
• C. $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$
• D. $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+xy=0$

Answer 1

The correct option is A $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$

$y=acos\left(logx\right)+bsin\left(logx\right)$

Differentiating w.r.t. x, we get,

$\frac{dy}{dx}=-asin\left(logx\right)\times \frac{1}{x}+bcosx\left(logx\right)\times \frac{1}{x}$

$x\frac{dy}{dx}=-asin\left(logx\right)+bcos\left(logx\right)$

again differentiating w.r.t. x

$x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.1=-acos\left(logx\right)\times \frac{1}{x}-bsin\left(logx\right)\times \frac{1}{x}$

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=-acos\left(logx\right)-bsin\left(logx\right)$

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=-\left[acos\right(logx)+bcos(logx\left)\right]$

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

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