Differential Equations · Homogeneous differential equations

Flawed question. Ill-posed: the solution is $\cos(y/x) = \log_e x$, which exists only for $e^{-1} \le x \le e$, so $y(e^{12})$ does not exist and $\alpha = \cos(\cdot) = 12$ is impossible; even with the formal $\alpha = 12$ ($r^2 = 2p^2 - 14$) only $p = \pm3, \pm4, \pm5$ give a circle (6 values), while the key 11 also counts $p$ with $r^2 < 0$; no minimal, evident change repairs both. It is shown for completeness and is left out of practice tests.