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Q.
The differential equation of all circles passing through the origin and having their centres on the $X$-axis is
Differential Equations
Solution:
The equation of all circles through the origin and having their centres on the $X$-axis is
$ (x-g)^2+(y-0)^2 =g^2 $
$\Rightarrow x^2+y^2-2 g x =0 .....$(i)
On differentiating equation (i) w.r.t. $x$, we get
$2 x+2 y \frac{d y}{d x}-2 g =0 $
$\Rightarrow 2 g =2\left(x+y \frac{d y}{d x}\right)$
On putting the value of $2 g$ in Eq. (i), we get
$ x^2+y^2-2\left(x+y \frac{d y}{d x}\right) x=0$
$ \Rightarrow y^2-x^2-2 x y \frac{d y}{d x}=0 $
$ \Rightarrow y^2=x^2+2 x y \frac{d y}{d x} $
