Form the differential equation of the family of all circles touching the $y-axis$ at the origin.

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Form the differential equation of the family of all circles touching the $y-axis$ at the origin.

Equation of the family of all circles touching the $y$ -axis at the origin can be represented by $(x-a)^{2}+y^{2}=a^{2}$ , where $a$ is an arbitrary constants.
$(x-a)^{2}+y^{2}=a^{2}(1)$
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 $a$ in equation (1)
$\begin{array}{l} \left(y \frac{d y}{d x}\right)^{2}+y^{2}=\left(x+y \frac{d y}{d x}\right)^{2} \\ \left(y \frac{d y}{d x}\right)^{2}+y^{2}=x^{2}+x y \frac{d y}{d x}+\left(y \frac{d y}{d x}\right)^{2} \end{array}$
Rearranging the above equation
$\mathrm{y}^{2}-\mathrm{x}^{2}-\mathrm{xy} \frac{\mathrm{dy}}{\mathrm{dx}}=0$
This is the required differential equation.