Differential Equation defining Confocal Conics

Theorem
Consider the equation:
 * $(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$

where $a^2 > b^2$ and $-\lambda < a^2$.

defining the set of confocal conics whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.

The differential equation defining these confocal conics is:
 * $x y \paren {\paren {y'}^2 - 1} + \paren {x^2 - y^2 - a^2 + b^2} y' = 0$

Proof
Then we have:

and similarly:

Eliminating $\lambda$: