Find $\frac{dy}{dx}$ when $y=(\log x{)}^x+x^{\log x}$

 If $y=5\cos x-3\sin x,$ prove that $\frac{d^{2}y}{d{x}^2}+y=0$.

If $y=a\cos \left(\log x\right)+b\sin \left(\log x\right)$, prove that $x^2{y}_2+x{y}_1+y=0$