Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.

Explanation

Equation of the circles which pass through the origin and whose centres lie on y-axis (say `(0,k)`) is,

`x^2+(y-k)^2=k^2 `

`implies k={x^2+y^2}/{2y}`

Differentiating it with respect to `x`, we get

`{2y(2x+2y{dy}/{dx})-2(x^2+y^2) {dy}/{dx}}/{4y^2}=0`

`implies (x^2-y^2) {dy}/{dx}-2xy=0`

This is the required differential equation.