To find a differential equation whose solution satisfies the given relation, let's first analyze the given equation. We are given:

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

Let's rewrite this to understand the structure of the equation.

1. Complete the square for both $x$ and $y$ terms:

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

   Completing the square for $x$ and $y$:

   $(x - A)^2 + (y - B)^2 = A^2 + B^2 - C$

   This equation represents a circle with center $(A, B)$ and radius $\sqrt{A^2 + B^2 - C}$.

2. Determine the Differential Equation:

   To find a differential equation corresponding to this solution, differentiate both sides of the equation implicitly with respect to $x$.

   Since $(x - A)^2 + (y - B)^2 = A^2 + B^2 - C$, we differentiate each side with respect to $x$:

   $2(x - A) + 2(y - B) \frac{dy}{dx} = 0$

   Simplify by dividing by 2:

   $(x - A) + (y - B) \frac{dy}{dx} = 0$

3. Rewrite the Resulting Differential Equation:

   Rearranging terms, we obtain:

   $\frac{dy}{dx} = -\frac{x - A}{y - B}$

This is the required differential equation. The solution to this differential equation will be a family of circles centered at $(A, B)$ with radius $\sqrt{A^2 + B^2 - C}$, under the condition that $A^2 + B^2 > C$.

Summary

The differential equation is:

$\frac{dy}{dx} = -\frac{x - A}{y - B}$

Would you like further details or have any specific questions?

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Related Questions

1. How do you find the general solution to this differential equation?
2. What are the possible values of $A$, $B$, and $C$ if the radius is zero?
3. How does the differential equation change if we vary $C$?
4. What happens if $A^2 + B^2 = C$?
5. Can this differential equation represent any other geometric shapes?

Tip

When dealing with equations of circles, completing the square can often simplify the process of finding a related differential equation.