1. **Stating the problem:** We are given a differential equation of the form $$P(y) \, dx + Q(x) \, dy = 0$$ where $P(y)$ is a function of $y$ only and $Q(x)$ is a function of $x$ only. We need to determine under what conditions this equation is exact. 2. **Recall the definition of an exact differential equation:** A differential equation $$M(x,y) \, dx + N(x,y) \, dy = 0$$ is exact if there exists a function $F(x,y)$ such that $$\frac{\partial F}{\partial x} = M(x,y)$$ and $$\frac{\partial F}{\partial y} = N(x,y).$$ This implies the condition $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$$ 3. **Apply the exactness condition to our equation:** Here, $$M(x,y) = P(y)$$ and $$N(x,y) = Q(x).$$ Since $P$ depends only on $y$ and $Q$ depends only on $x$, we have: $$\frac{\partial M}{\partial y} = \frac{dP}{dy}$$ and $$\frac{\partial N}{\partial x} = \frac{dQ}{dx}.$$ 4. **Exactness condition becomes:** $$\frac{dP}{dy} = \frac{dQ}{dx}.$$ Since the left side depends only on $y$ and the right side depends only on $x$, for this equality to hold for all $x,y$, both sides must be equal to the same constant, say $A$: $$\frac{dP}{dy} = A \quad \text{and} \quad \frac{dQ}{dx} = A.$$ 5. **Integrate both sides:** $$P(y) = Ay + B$$ $$Q(x) = Ax + C$$ where $B$ and $C$ are constants of integration. 6. **Conclusion:** The differential equation is exact if and only if $P(y)$ and $Q(x)$ are linear functions with the same slope $A$. **Answer:** The correct choice is: "is exact only if $P(y) = Ay + B$ and $Q(x) = Ax + C$ where $A,B,C$ are constants."