Question

If y=acos(logx)+bsin(logx), show that $x^2{y}_2+x{y}_1+y=0$

Answer 1

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

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

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

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

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

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

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