Ode containing triangle trigonometry

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What has the homogeneous differential equation $$ \frac{dy}{dx}= \frac{x^2+3 y^2}{2 x y}$$

its solution written on blackboard ( Germany, 1894 )

$$ x^2 +y^2 - C x^3 = 0 $$

to do with Area of the triangle using Sine or Cosine Rule in Trigonometry ?

$$ Area = \frac {c^2 \sin A \sin B}{2\sin C}$$

What do $(x,y)$ of that equation represent in the curve below $ (C=1)?$

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