Answer
The differential equation
$$
\frac{d y}{d x}=-\frac{a x-b y}{b x-c y}
$$
is not exact.
Work Step by Step
$$
\frac{d y}{d x}=-\frac{a x-b y}{b x-c y}
$$
this equation can be written as
$$
\left(a x-b y\right) d x+\left(b x-c y\right) d y=0
$$
Comparing this Equation with the differential form:
$$
M(x,y) d x+N(x,y) d y=0
$$
we observe that
$$
\begin{aligned}M(x,y) &=\left(a x-b y\right)
\\ N(x,y) &=\left(b x-c y\right) \end{aligned}
$$
By calculating $M_{y}$ and $N_{x}$ , we find that
$$
\begin{aligned}M_{y}(x,y) &=-b
\\ N_{x}(x,y) &=b \end{aligned}
$$
we obtain that
$$
M_{y}(x, y) \neq N_{x}(x, y)
$$
so the given equation is not exact.