Question

The differential equation of all circles with a centre at the origin is?

Answer 1

Determine the differential equation.

The equation of a circle centered at the origin is $x^2+y^2=r^2$.

Differentiate the function with respect to $x$:

$\begin{array}{rcl}\frac{d}{dx}\left(x^2+y^2\right)& =& \frac{d}{dx}\left(r^2\right)\\ & \Rightarrow & 2x+2y\frac{dy}{dx}=0\\ & \Rightarrow & 2xdx+2ydy=0\end{array}$

So, the required differential equation of circle is $\begin{array}{rcl}2xdx+2ydy& =& 0\mathrm{or}xdx+ydy=0\end{array}$.