# Solution required

• Mar 9th 2008, 04:02 AM
Altair
Solution required
Q.Find the differential equation of all the circles that pass through origin.

A. $(x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2)$
• Mar 9th 2008, 04:36 AM
mr fantastic
Quote:

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).
• Mar 10th 2008, 10:49 AM
Altair
Need help with substitution. It gets too complicated at the end.
• Mar 10th 2008, 12:58 PM
mr fantastic
Quote:

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.