If $y = 2\cos(\log x) - 5\sin(\log x)$, what is $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx}$?

Explanation

1. 01

   $x\frac{dy}{dx} = -2\sin(\log x) - 5\cos(\log x)$.

2. 02

   Differentiating that: $x\frac{d^2y}{dx^2} + \frac{dy}{dx} = \frac{-2\cos(\log x) + 5\sin(\log x)}{x} = -\frac{y}{x}$.

3. 03

   Multiplying by $x$: $x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} = -y$.