The solution of the differential equation $(\cot y) dx=x ~dy$ is :

$x=C \sec y$

cot y dx = x dy

so  $\frac{1}{x} dx = \frac{1}{\cot y}dy$

⇒  $\int \frac{1}{x}dx = \int \tan y dy$

integrating both sides w.r.t x

⇒  $\log x = \log \sec y + \log k$

taking constant in terms of log for case of calculation

so $\log x = \log k \sec y$

⇒  $x = k \sec y$

let arbitrary constant

k = C

x = C sec y