Question

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

Answer 1

The system of circles touching $y$ axis at origin will have centres on $x$ axis. Let $(a,0)$ be the centre of the circle. Then the radius of the circle should be $a$ units, since the circle should touch $y$ axis at origin.

Equation of a circle with centre at $(a,0)$ and radius $a$ is

$x^2+y^2-2ax=0$

The above equation represents the family of circles touching $y$ axis at origin. Here $a$ is an arbitrary constant.

In order to find the differential equation of system of circles touching $y$ axis at origin, eliminate the arbitrary constant from equation.

Differentiating equation with respect to $x$, we get

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

Replacing $2a$ of the above equation, we get

$x^2+y^2-2(x+y\frac{dy}{dx})x$

Which is $x^2-y^2+2xy\frac{dy}{dx}=0$