Form the differential equation of all circle touching the x-axis at the origin and centre on the y-axis.

Explanation

If center of the circle is (0,a) then its radius is `|a|`.
`therefore` equation of such family of circles is given by
`x^(2)+(y-a)^(2)=a^(2)`
or `x^(2)+y^(2)-2ay=0`……………..(1)
Differentiating w.r.t. x, we get
`2x+2y(dy)/(dx)=2a(dy)/(dx)`...............(2)
Form (1), `2a=(x^(2)+y^(2))/(y)`
Putting, the value of 2a in (2), we get
`2x+2y(dy)/(dx)=(x^(2)+y^(2))/(y)(dy)/(dx)`
or `(x^(2)-y^(2))(dy)/(dx)=2xy,` which is required differential equation,