Q.Find the differential equation of all the circles that pass through origin.
If the circle passes through the origin then .
Then the equation of the circle becomes
Differentiate (1) with respect to x:
Differentiate (2) with respect to x:
Clearly k needs to be eliminated from (3).
(1) and (2) can be solved simultaneously to get k and h in terms of x, y and y'. You want k. Substitute the result into (3).