1. **Problem:** Eliminate the arbitrary constant $C$ from the equation $$y = Cx + C^2 + 1$$ 2. **Step 1: Differentiate both sides with respect to $x$** $$\frac{dy}{dx} = C + 0 + 0 = C$$ 3. **Step 2: Express $C$ from the derivative:** $$C = \frac{dy}{dx}$$ 4. **Step 3: Substitute $C$ back into the original equation:** $$y = \left(\frac{dy}{dx}\right) x + \left(\frac{dy}{dx}\right)^2 + 1$$ 5. **Step 4: Rearrange to form a differential equation:** $$y = x \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 + 1$$ 6. **Step 5: Write the final implicit differential equation:** $$y - x \frac{dy}{dx} - \left(\frac{dy}{dx}\right)^2 = 1$$ This equation no longer contains the arbitrary constant $C$. **Explanation:** We used differentiation to find an expression for the constant $C$ in terms of $x$ and $y$. Substituting back eliminated $C$, giving a differential equation involving only $x$, $y$, and derivatives of $y$.