Question

Write the order of the differential equation of the family of circles touching X-axis at the origin.

Answer 1

The equation of the family of circles touching x-axis at the origin is \[\left( x - 0 \right)^2 + \left( y - a \right)^2 = a^2 \]
\[ x^2 + y^2 - 2ay = 0 . . . . . \left( 1 \right)\]
Here, a is the parameter . 
Since this equation contains only one arbitary constant, we differentiate it only once . 
Differentiating with respect to x, we get
\[2x + 2y\frac{dy}{dx} - 2a\frac{dy}{dx} = 0\]
\[a = \frac{x + y\left( \frac{dy}{dx} \right)}{\frac{dy}{dx}} . . . . . \left( 2 \right)\]
Putting the value of a from (2) in (1), we get 
\[ x^2 + y^2 = 2y\left\{ \frac{x + y\left( \frac{dy}{dx} \right)}{\frac{dy}{dx}} \right\}\]
\[\left( x^2 - y^2 \right)\frac{dy}{dx} = 2xy\]
So, this is the required differential equation . 
Here, order of the differential equation is 1 .