1. ## Solution required

Q.Find the differential equation of all the circles that pass through origin.

A. $(x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2)$

2. Originally Posted by Altair
Q.Find the differential equation of all the circles that pass through origin.

A. $(x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2)$
Well, I have a plan but you'll need to put in the details.

If the circle $(x - h)^2 + (y - k)^2 = r^2$ passes through the origin then $h^2 + k^2 = r^2$.

Then the equation of the circle becomes

$x^2 - 2xh + y^2 - 2yk = 0$ .... (1)

Differentiate (1) with respect to x:

$2x - 2h + 2y \, y' - 2k \, y' = 0$ .... (2)

Differentiate (2) with respect to x:

$2 + 2 (y')^2 + 2 y \, y'' - 2k \, y'' = 0$ .... (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).

3. Need help with substitution. It gets too complicated at the end.

4. Originally Posted by Altair
Need help with substitution. It gets too complicated at the end.
I'll take a closer look when I have a chance, unless someone else beats me to it.