I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic!
I know that confocal ellipses and hyperbolas centered at the origin follow the equation:
$\frac{x^2}{a^2} + \frac{y^2}{a^2 - f^2} = 1 ~~ (1)$
By differentiating with respect to $x$ once, and getting rid of our constant $a$, we can arrive at the following first order differential equation:
$x y (\frac{dy}{dx})^2 + (x^2 - y^2 - f^2)\frac{dy}{dx} - xy = 0 ~~ (2)$
This equation represents both, ellipses and hyperbolas centered at the origin with foci at $f$ and $-f$. I am clear with everything so far.
The problem lies when I try to go back from the differential equation to the original representation.
Now, I know that the solution to the differential equation must be equation $(1)$, but I am having a very hard time solving equation $(2)$.
Since this can be seen as a second degree polynomial on $\frac{dy}{dx}$, I thought about just using the quadratic formula, but I get a horrible expression:
$\frac{dy}{dx} = \frac{-(x^2 + y^2 - f^2) \pm \sqrt{(x^2 - y^2 - f^2)^2 + 4x^2y^2}}{2xy} = \frac{-(x^2 + y^2 - f^2) \pm \sqrt{x^4 - 2x^2y^2 - 2x^2f^2 + y^4 +2y^2 f^2 + f^4 + 4x^2y^2}}{2xy} $
$\frac{dy}{dx} = \frac{-(x^2 + y^2 - f^2) \pm \sqrt{x^4 + 2x^2y^2 - 2x^2f^2 + y^4 +2y^2 f^2 + f^4 }}{2xy} = \frac{-(x^2 + y^2 - f^2) \pm \sqrt{(x^2 + y^2)^2 - 2f^2 (x^2 - y^2) + f^4}}{2xy} $
Now, I really have no idea how to integrate this...
I know that the solution to this equation must be equation $(1)$, but I have no idea how.
Help would be greatly appreciated!