Equation of circles of radius $r$ :-
$$ {( x-a)}^2 +{(y - b)^2} = r^2 $$
Then I treated $a,b,r$ as arbitrary constants .
And differentiated the equation thrice .
which gave the wrong result
As in the solution in the book equation was differentiated twice .
So why do we not treat $r$ as arbitrary constant and differentiate it twice ?
My question is very different from "Find the differential equation of all circles of radius a" as this question is aking what to do whereas i am asking why do not we take radius as arbitrary.
The question should be interpreted as “find the differential equation for all circles of a fixed (given) radius $r$”. If you treat $r$ as arbitrary, you get the differential equations for all circles, no matter what radius.