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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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NatGeo video 2 min 32 sec onwards

Narasimham
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  • I shall follow this question. Bizarre, isn't it ? – Claude Leibovici Sep 24 '21 at 14:10
  • Yup. Hope you would enjoy it. – Narasimham Sep 24 '21 at 14:11
  • $dx=2y d\varphi$, but I don't see how it is connected with area. – Ivan Kaznacheyeu Sep 24 '21 at 14:31
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    Connected: https://math.stackexchange.com/q/2308674 – Jean Marie Sep 24 '21 at 14:35
  • A bit of context to avoid to visiting YouTube: In the clip (from National Geographic's 2017 series "Genius", about Albert Einstein), a young Einstein shows that he's been paying (enough) attention during a math lecture by effortlessly solving the diff eqn, which had been posed by the professor while Einstein was seemingly lost in thought. ... To the question at hand: The prof prefaced the eqn by having his students recall/recite the Laws of Cosines and Sines. This suggests the trig connection, but the exact connection isn't clear. (The professor's words had become distant for dramatic effect.) – Blue Sep 24 '21 at 16:41
  • Reputed channel National Geographic had 45 secs for an episode like this...must have been after due factual researches. – Narasimham Sep 24 '21 at 23:02

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