Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Let the centre of the circle on y – axis be (0,b).

We know that the differential equation of the family of circles with centre at (0, b) and radius 3 is: x^{2} + (y- b)^{2} = 3^{2}

⇒ x^{2} + (y- b)^{2} = 9 ----(1)

Now, differentiating both sides w.r.t. x, we get,

2x + 2(y – b).y’ = 0

⇒ (y – b). y’ = x

⇒ y – b =

Thus, substituting the value of ( y – b) in equation (1), we get,

⇒ x^{2}((y’)^{2} + 1) = 9(y’)^{2}

⇒ (x^{2} – 9)(y’)^{2} + x^{2} = 0

Therefore, the required differential equation is (x^{2} – 9)(y’)^{2} + x^{2} = 0

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