Answer
The differential equation
$$
\left(e^{x} \sin y+3 y\right) d x-\left(3 x-e^{x} \sin y\right) d y=0
$$
is not exact.
Work Step by Step
$$
\left(e^{x} \sin y+3 y\right) d x-\left(3 x-e^{x} \sin y\right) d y=0
\quad \quad (i)
$$
Comparing this Equation with the differential form:
$$
M(x,y) d x+N(x,y) d y=0
$$
we observe that
$$
\begin{aligned}M(x,y) &=\left(e^{x} \sin y+3 y\right)
\\ N(x,y) &=-\left(3 x-e^{x} \sin y\right) \end{aligned}
$$
By calculating $M_{y}$ and $N_{x}$ , we find that
$$
\begin{aligned} M_{y}(x, y) &=e^{x} \cos y+3
\\ N_{x}(x, y) & =-3+e^{x} \sin y \end{aligned}
$$
we obtain that
$$
M_{y}(x, y) \neq N_{x}(x, y)
$$
so the given equation is not exact.
