Partial differential equation solution?

Questions
,
1

I am trying to find the solution of the equation x cos[y] dx+ sin x dy=0 in Mathematica, but couldn’t find the solution. Professionals, Kindly help me out with it.

DSolve[y'[x] == Sin[x]/(x Cos[y[x]]), y[x], x]
2

$Version

(* "12.3.0 for Microsoft Windows (64-bit) (May 10, 2021)" *)

DSolve[y'[x] == Sin[x]/(x Cos[y[x]]), y[x], x]

 (*{{y[x] -> ArcSin[C[1] + SinIntegral[x]]}}*)
3

Thank you, Sir.

4

Ok, but I think it should be the other way round because the question was x cos[y] dx+ sin x dy=0 :

DSolve[y'[x] == (x Cos[y[x]])/Sin[x], y[x], x]