The differential equation of all circles, passing through the origin and having their centres on the X-axis,…
The differential equation of all circles, passing through the origin and having their centres on the X-axis, is
$y^2=x^2+x y \frac{\mathrm{d} y}{\mathrm{~d} x}$
$x^2=y^2+2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}$
$y^2=x^2+2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}$
$x^2=y^2-x y \frac{\mathrm{d} y}{\mathrm{~d} x}$
Solution
The system of circles which passes through origin and whose centre lies on $\mathrm{X}$-axis is $x^2+y^2-2 b x=0... (i)$
Differentiating w.r.t $x$, we get
$2 x+2 y \frac{\mathrm{d} y}{\mathrm{~d} x}=2 \mathrm{~b}... (ii)$
Substituting (ii) in (i), we get
$\begin{aligned}
& x^2+y^2-2 x^2-2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}=0 \\
& \Rightarrow y^2-x^2-2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}=0 \\
& \Rightarrow y^2=x^2+2 x y \frac{\mathrm{d} y}{\mathrm{~d} x}
\end{aligned}$