Sol. The equation of the family of circles of radius r is (x –a) 2 + (y –b) 2 = r 2 ... (i)

where a and b are arbitrary constants, we differentiate it two times w.r.t x and the differential equation will be of second order.

Differentiating (i) w.r.t. x and the differential equation will be of second order.

Differentiating (i) w.r.t x, we get

2 (x – a) + 2(y – b) = 0

⇒ (x – a) + (y – b) = 0 ... (ii)

Differentiating (ii) w.r.t. x, we get

1 + (y –b) + = 0 ... (iii)

From (iii), we have

y –b = – ... (iv)

Putting the value of (y –b) in (ii), we obtain

x – a= ...(v)

Substituting the values of (x – a) and (y – b) in (i), we get

+ = r 2 ⇒ = r

2

This is the required differential equation.