Form the differential equation of the family of circles having centers on the $x-axis$ and radius unity.

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Form the differential equation of the family of circles having centers on the $x-axis$ and radius unity.

Equation of the family of circles having centers on the $x$ -axis and radius unity can be represented by $(x-a)^{2}+(y)^{2}=1$ , where $a$ is an arbitrary constants.
$(x-a)^{2}+y^{2}=1(1)$
Differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l} 2(x-a)+2(y) \frac{d y}{d x}=0 \\ x-a+y \frac{d y}{d x}=0 \\ a=x+y \frac{d y}{d x} \end{array}$
Substituting the value of $\mathrm{a}$ in equation (1)
$\begin{array}{l} \left(x-x-y \frac{d y}{d x}\right)^{2}+y^{2}=1 \\ \left(y \frac{d y}{d x}\right)^{2}+y^{2}=1 \end{array}$
This is the required differential equation.