Answer
The equation
\begin{equation}
(x \ln y+x y) d x+(y \ln x+x y) d y=0 ; \quad x>0, \quad y>0
\end{equation}
is not exact.
Work Step by Step
\begin{equation}
(x \ln y+x y) d x+(y \ln x+x y) d y=0 ; \quad x>0, \quad y>0 \quad (i)
\end{equation}
Comparing this Equation with the differential form:
$$
M(x,y) d x+N(x,y) d y=0
$$
we observe that
$$
\begin{aligned}M(x,y) &=(x \ln y+x y)
\\ N(x,y) &=(y \ln x+x y) \end{aligned}
$$
By calculating $M_{y}$ and $N_{x}$ , we find that
\begin{aligned}M_{y}(x,y) &=\frac{x}{y}+x
\\ N_{x}(x,y) &=\frac{y}{x} +y \end{aligned}
we obtain that
$$
M_{y}(x, y) \neq N_{x}(x, y)
$$
so the given equation is not exact.