The differential equation of the family of circles touching $y$-axis at the origin is
The differential equation of the family of circles touching $y$-axis at the origin is
$x^2-y^2-2 x y \frac{d y}{d x}=0$
$x^2-y^2+2 x y \frac{d y}{d x}=0$
$x^2+y^2-2 x y \frac{d y}{d x}=0$
$x^2+y^2+2 x y \frac{d y}{d x}=0$
Solution
Since circles touch $\mathrm{Y}$ axis at origin, the centres of the circles lie on $\mathrm{X}$ axis.
Let centre be $(\mathrm{h}, 0)$ and radius $=\mathrm{h}$
$\therefore(\mathrm{x}-\mathrm{h})^2+(\mathrm{y}-0)^2=\mathrm{h}^2 \Rightarrow \mathrm{x}^2-2 \mathrm{hx}+\mathrm{y}^2=0$
Differentiating w.r.t. $\mathrm{x}$, we get
$2 x-2 h+2 y \frac{d y}{d x}=0 \quad \Rightarrow x+y \frac{d y}{d x}=h$
Substituting value of $h$ in eq. (1), we get
$\begin{aligned}
& x^2-2\left(x+y \frac{d y}{d x}\right) x+y^2=0 \\
& \therefore x^2-2 x^2-2 x y \frac{d y}{d x}+y^2=0 \quad \Rightarrow x^2-y^2+2 x y \frac{d y}{d x}=0
\end{aligned}$