Take $x, y > 0$. Which of these differential equations is homogeneous?

Explanation

1. 01

   $\log x - \log y = -\log\frac{y}{x}$, so $\frac{y}{x} + \log x - \log y = v - \log v$, a function of $v = \frac{y}{x}$ alone.

2. 02

   $\log x + \log y$, $\log x$ and $\log y$ each change by a multiple of $\log\lambda$ when $x, y$ become $\lambda x, \lambda y$, so the others fail.