Question

Obtain the differential equation by eliminating arbitrary constants $A,B$ from the equation-
$y=A\cos \left(\log x\right)+B\sin \left(\log x\right)$.

Answer 1

Given, $y=A\cos \left(\log x\right)+B\sin \left(\log x\right)$
Differentiating w.r to $x$, we get
$\frac{dy}{dx}=-A\sin \left(\log x\right)\times \frac{1}{x}+B\cos \left(\log x\right)\times \frac{1}{x}$
$x\frac{dy}{dx}=-A\sin \left(\log x\right)+B\cos \left(\log x\right)$
Again differentiating, we get
$x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-A\cos \left(\log x\right)\times \frac{1}{x}-B\sin \left(\log x\right)\times \frac{1}{x}$
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=-[A\cos \left(\log x\right)+B\sin \left(\log x\right)]$
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=-y$
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$