Question

Find the order of the differential equation of the family of all the circles with their centres at the origin.

Answer 1

$\to$ For the family of circle general equation is $(x-h{)}^2+(y-k{)}^2=r^2$

Here, $(h,k)=(0,0)$ as their centres are at origin.

$\cdot$ For circle $1$ lets assume radius$r_1$ and we know centre is $(0,0).$

$\to$ Let us assume another circle $2$ with radius $r_2$ and which has center $(0,0)$

$\cdot$ Similarly many concentric circles will be formed.

(Refer the image)

For circle $1$ , $x^2+y^2=r_1^2$

For circle $2,x^2+y^2=r_2^2$

$\to$ The differential equation we will find after differentiating any above equation w.r.t. $x$ so we get :

$\therefore x^2+y^2=r_1^2$

$\therefore 2x+2y=\frac{dy}{dx}=0$

$\therefore \frac{dy}{dx}=\left(\frac{-x}{y}\right)$

Therefore, the order of differential equation is $1.$