1. **State the problem:** We are given the implicit equation $$x^2 + y^2 = A e^{B x}$$ and asked to find the resulting differential equation by differentiating with respect to $x$. 2. **Recall the formula and rules:** To find the differential equation, we differentiate both sides with respect to $x$. Remember that $y$ is a function of $x$, so use implicit differentiation and the chain rule: $$\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$$. 3. **Differentiate both sides:** $$\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(A e^{B x})$$ $$2x + 2y \frac{dy}{dx} = A B e^{B x}$$ 4. **Express $A e^{B x}$ from the original equation:** $$A e^{B x} = x^2 + y^2$$ 5. **Substitute back to eliminate $A e^{B x}$:** $$2x + 2y \frac{dy}{dx} = B (x^2 + y^2)$$ 6. **Solve for $\frac{dy}{dx}$:** $$2y \frac{dy}{dx} = B (x^2 + y^2) - 2x$$ $$\frac{dy}{dx} = \frac{B (x^2 + y^2) - 2x}{2y}$$ **Final answer:** The resulting differential equation is $$\boxed{\frac{dy}{dx} = \frac{B (x^2 + y^2) - 2x}{2y}}$$