Question

Obtain the differential equation of the family of circles touching the y-axis at the origin.

Answer 1

Let centre be $(a,0)$

So, radius will be ${}^{\prime }{a}^{\prime }$ since circle is touching y-axis.

Therefore, equation will be

${(x-a)}^2+y^2=a^2$

differentiating it, we get

$2(x-a)+2y\frac{dy}{dx}=0a=x+y\frac{dy}{dx}$

Put it in the original equation , we get

$y^2{\left(\frac{dy}{dx}\right)}^2+y^2={(x+y\frac{dy}{dx})}^2{y}^2-x^2=2xy\frac{dy}{dx}$

or$\frac{y}{x}-\frac{x}{y}=2\frac{dy}{dx}$