Question

Form the differential equation of the circles touching the $x$-axis at origin.

Answer 1

$x^2+y^2+2fy=\mathrm{0....}\left(1\right)$ is family of circles touching x axis for some $f$

Differentiate wrt $x$

$2x+2y{y}^{\prime }+2f{y}^{\prime }=0$

$x+y{y}^{\prime }=-f{y}^{\prime }$

$f=-\frac{x+y{y}^{\prime }}{y^{\prime }}$

Substitute the above value of $fin\left(1\right)$

$x^2+y^2+2\times -\frac{x+y{y}^{\prime }}{y^{\prime }}\times y=0$

$(x^2+y^2)y^{\prime }-2x-2y{y}^{\prime }=0$ is required differential equation