Find the differential equation of all the circles which pass through the origin and whose centers lie on the x - axis.
Any circle with centre at (h, k) and radius r is given by,
(x – h)2 + (y – k)2 = r2
Here centre is on x - axis, so k = 0
So, we have the equation of circle as, (x – h)2 + y2 = r2
Further it is given that circle passes through origin (0,0) therefore origin must satisfy equation of circle. So, we get,
0 + h2 = r2
So, the equation of circle is (x – h)2 + y2 = h2
⇒ x2 – 2hx + y2 = 0
⇒ x2 + y2 = 2hx
Now, differentiating it with respect to x we get,
Hence, the required differential equation is