Integer parameter in an implicit radical solution

The particular solution on the interval \(x>0\) of the differential equation \((x+\sqrt{xy})y'=y\) satisfies \(y(1)=1\).

The implicit equation this solution satisfies is \(\sqrt{y}(\ln y+k)=k\sqrt{x}\), for some integer number \(k\).

Find the value of \(k\).