# Solution required

• Mar 9th 2008, 03:02 AM
Altair
Solution required
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) \$
• Mar 9th 2008, 03:36 AM
mr fantastic
Quote:

Originally Posted by Altair
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).
• Mar 10th 2008, 09:49 AM
Altair
Need help with substitution. It gets too complicated at the end.
• Mar 10th 2008, 11:58 AM
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.