Question

Prove that $y=acos\left(logx\right)+bsin\left(logx\right)$ is the solution of $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0.$

Answer 1

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

$\frac{dy}{dx}=-a\sin \left(\log x\right).\frac{1}{x}+b\cos \left(\log x\right).\frac{1}{x}$

$\frac{dy}{dx}=\frac{1}{x}\left(b\cos \right(\log x)-a\sin (\log x\left)\right)$

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

$x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-b\sin \left(\log x\right).\frac{1}{x}-a\cos \left(\log x\right).\frac{1}{x}$

$=-\frac{1}{x}\left(a\cos \right(\log x)+b\sin (\log x\left)\right)$

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

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