Obtain the differential equation of the family of circles :
having radius 3 and centre on y-axis.

Answer

To obtain the differential equation of the family of circles having a radius of 3 and centers on the y-axis, we can follow these steps:

### Step 1: Write the equation of the circle
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) \) is the center of the circle and \( r \) is the radius. Since the center lies on the y-axis, we have \( h = 0 \) and \( k \) can be any value. Given that the radius \( r = 3 \), the equation becomes:
\[ (x - 0)^2 + (y - k)^2 = 3^2 \]
which simplifies to:
\[ x^2 + (y - k)^2 = 9 \]

### Step 2: Differentiate the equation
Next, we differentiate the equation with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}((y - k)^2) = \frac{d}{dx}(9) \]
This gives:
\[ 2x + 2(y - k)\frac{dy}{dx} = 0 \]

### Step 3: Solve for \( y - k \)
Rearranging the differentiated equation, we find:
\[ 2(y - k)\frac{dy}{dx} = -2x \]
Dividing both sides by 2:
\[ (y - k)\frac{dy}{dx} = -x \]
Now, we can express \( y - k \) in terms of \( x \) and \( \frac{dy}{dx} \):
\[ y - k = -\frac{x}{\frac{dy}{dx}} \]

### Step 4: Substitute back into the original equation
Now we substitute \( y - k \) back into the original circle equation:
\[ x^2 + \left(-\frac{x}{\frac{dy}{dx}}\right)^2 = 9 \]
This simplifies to:
\[ x^2 + \frac{x^2}{\left(\frac{dy}{dx}\right)^2} = 9 \]

### Step 5: Multiply through by \( \left(\frac{dy}{dx}\right)^2 \)
To eliminate the fraction, multiply the entire equation by \( \left(\frac{dy}{dx}\right)^2 \):
\[ x^2\left(\frac{dy}{dx}\right)^2 + x^2 = 9\left(\frac{dy}{dx}\right)^2 \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ x^2\left(\frac{dy}{dx}\right)^2 - 9\left(\frac{dy}{dx}\right)^2 + x^2 = 0 \]
Factoring out \( \left(\frac{dy}{dx}\right)^2 \):
\[ \left(\frac{dy}{dx}\right)^2(x^2 - 9) + x^2 = 0 \]

### Final Step: Write the final differential equation
Thus, the final form of the differential equation is:
\[ x^2\left(\frac{dy}{dx}\right)^2 + x^2 - 9\left(\frac{dy}{dx}\right)^2 = 0 \]