To find a differential equation whose solution is given by:

$x^2 + y^2 - 2Ax - 2By + C = 0,$

we can proceed by differentiating this implicit equation with respect to $x$ and then eliminate the constants $A$, $B$, and $C$ to obtain a differential equation in terms of $x$, $y$, and $\frac{dy}{dx}$.

Step 1: Differentiate with respect to $x$

The given equation is: $x^2 + y^2 - 2Ax - 2By + C = 0.$

Differentiating with respect to $x$, we obtain: $2x + 2y \frac{dy}{dx} - 2A - 2B \frac{dy}{dx} = 0.$

Simplify this to: $(x - A) + (y - B) \frac{dy}{dx} = 0.$

Step 2: Differentiate again to eliminate $A$ and $B$

Now, differentiate once more with respect to $x$: $1 + \frac{dy}{dx} \cdot \frac{dy}{dx} + (y - B) \frac{d^2y}{dx^2} = 0.$

This becomes: $1 + \left( \frac{dy}{dx} \right)^2 + (y - B) \frac{d^2y}{dx^2} = 0.$

Since $B$ was arbitrary, express $B$ in terms of $x$, $y$, and their derivatives to obtain the final differential equation