Find the differential equation of the family of concentric circles `x^2+y^2=a^2`.

Answer

To find the differential equation of the family of concentric circles given by the equation \( x^2 + y^2 = a^2 \), we will follow these steps:

### Step 1: Differentiate the equation
We start with the equation of the circle:
\[ x^2 + y^2 = a^2 \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2) \]
This gives us:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 2: Rearranging the equation
Now, we can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} = -2x \]
Dividing both sides by \( 2y \):
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Step 3: Eliminate the parameter \( a \)
Since \( a \) is a constant and we are looking for a differential equation, we can eliminate \( a \) from our equations. We know from the original equation that:
\[ a^2 = x^2 + y^2 \]
Thus, we can express \( a \) in terms of \( x \) and \( y \):
\[ a = \sqrt{x^2 + y^2} \]
However, since we already have \( \frac{dy}{dx} = -\frac{x}{y} \), we can use this to express the relationship between \( x \) and \( y \) without needing \( a \).

### Final Differential Equation
The final differential equation representing the family of concentric circles is:
\[ \frac{dy}{dx} + \frac{x}{y} = 0 \]