1. Stating the problem: Given $$y = \cot(ax)$$ and the differential equation $$\frac{dy}{dx} + 4(1 + y^2) = 0$$, find the value of $$a$$. 2. Calculate $$\frac{dy}{dx}$$ from $$y = \cot(ax)$$. Recall that $$\frac{d}{dx}\cot(u) = -\csc^2(u) \cdot \frac{du}{dx}$$. Here, $$u = ax$$, so $$\frac{du}{dx} = a$$. Therefore, $$\frac{dy}{dx} = -a \csc^2(ax)$$. 3. Substitute $$y = \cot(ax)$$ and $$\frac{dy}{dx}$$ into the differential equation: $$-a \csc^2(ax) + 4(1 + \cot^2(ax)) = 0$$. 4. Use the identity: $$1 + \cot^2(\theta) = \csc^2(\theta)$$. So the equation becomes: $$-a \csc^2(ax) + 4 \csc^2(ax) = 0$$. 5. Factor out $$\csc^2(ax)$$: $$\csc^2(ax)(-a + 4) = 0$$. Since $$\csc^2(ax) \neq 0$$ for $$x$$ in the domain, we have: $$-a + 4 = 0$$. 6. Solve for $$a$$: $$a = 4$$. Final answer: $$a = 4$$. Thus, the correct choice is (c) 4.