Q.Find the differential equation of all the circles that pass through origin.
A. $\displaystyle (x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2) $
Well, I have a plan but you'll need to put in the details.
If the circle $\displaystyle (x - h)^2 + (y - k)^2 = r^2$ passes through the origin then $\displaystyle h^2 + k^2 = r^2$.
Then the equation of the circle becomes
$\displaystyle x^2 - 2xh + y^2 - 2yk = 0$ .... (1)
Differentiate (1) with respect to x:
$\displaystyle 2x - 2h + 2y \, y' - 2k \, y' = 0$ .... (2)
Differentiate (2) with respect to x:
$\displaystyle 2 + 2 (y')^2 + 2 y \, y'' - 2k \, y'' = 0$ .... (3)
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).