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Prove that conics are the integral curves of a differential equation

Using the Cartesian equation for conics of eccentricity e and center (0,0) to prove that these conics are the integral curves of the differential equation

    \[ y' = (e^2 - 1) \frac{x}{y}. \]


Proof. Incomplete.

One comment

  1. S says:

    If e=1, y=const, so the problem is for a hyperbola or ellipse. There we know x^2/a^2 + y^2/(a^2(1-e^2))=1. Taking a derivative, we get to the differential equation.

    We could also go the other way around, to find that solutions of the differential equation are as in the equation for the ellipse and hyperbola (where a^2 is contained in the integral constant C).

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