The differential equation of the system of circles touching the y-axis at the origin is

Calculation:

As we know that a circles touching y – axis at origin has the centre on x – axis.

So, the equation of circle touching y – axis at origin is: (x - a)² + y² = a²

⇒ x² + y² – 2ax = 0       ---(1)

So, by differentiating the above equation w.r.t x, we get

\(\Rightarrow 2x + 2y\frac{{dy}}{{dx}} - 2a = 0\)

\(\Rightarrow 2a = 2\;\left( {x + y\frac{{dy}}{{dx}}} \right)\)

So, by substituting the value of 2a in equation (1), we get

⇒ x2 + y2 – \(2x\;\left( {x + y\frac{{dy}}{{dx}}} \right)\)= 0

⇒ \( {2x^2 +2x y\frac{{dy}}{{dx}}} -x^2 -y^2 =0\)

\(\therefore {x^2} - {y^2} + 2xy\frac{{dy}}{dx} = 0\)