The center of the circle touching the y- axis at orgin lies on the x – axis.
Let (a,0) be the centre of the circle.
Thus, it touches the y – axis at orgin, its radius is a.
Now, the equation of the circle with centre (a,0) and radius (a) is
(x –a)2 – y2 = a2
⇒ x2 + y2 = 2ax
Now, differentiating both sides w.r.t. x , we get,
2x + 2yy’ = 2a
⇒ x + yy’ = a
Now, on substituting the value of a in the equation, we get,
x2 + y2 = 2(x + yy’)x
⇒ 2xyy’ + x2 = y2
Therefore, the required differential equation is 2xyy’ + x2 = y2 .
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