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Form the differential equation of all circles

Question:

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

Solution:

The equation of circles which pass through the origin and whose centre lies on the y-axis is given by,

(x – 0)2 + (y – a)2 = a2, where (0, a) is the centre

x2 + y2 + a2 – 2ay = a2

$\left(x^{2}-y^{2}\right) \frac{d y}{d x}-2 x y=0$

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