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In Genius Season 1 Episode 1 at 9:00, Young Albert Einstein defies his teacher by solving the equation on board and states

Natural log of constant multiplied by x equals natural log of one plus v squared. And since v equals y over x. That gives us the final function: x squared plus y squared minus c x cubed equals zero.

What is he talking about?

P.S.

The question was asked first on Movies/TV stackexchange and then on math stackexchange but it was suggested from the members that the question would fare well on physics stackexchange. However if someone considers this question unsuitable for the stackexchange let me know.

Edit:

I am perfectly aware what does the context translates into. What I am asking is whether the equation is known for something. Like "$c^2=a^2+b^2$" instantly make us recall of "Pythagoras Theorem". In the same sense, does the provided context above hold any significant meaning?

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  • $\begingroup$ sounds like an ODE where you substitute for $y/x$ $\endgroup$ – Spine Feast Jun 3 '17 at 22:55
  • $\begingroup$ Is there no special significance of this equation? $\endgroup$ – mathnoob123 Jun 3 '17 at 22:56
  • $\begingroup$ It describes a curve like this wolframalpha.com/input/?i=x%5E2+%2B+y%5E2+-+3x%5E3+%3D+0 ... might be the solution of an ODE arising from some sort of variational problem. You could work backwards from the logarithm equation because presumably they arise from integrating $dx/x = 2v dv/(1+v^2)$ $\endgroup$ – Spine Feast Jun 3 '17 at 23:05
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What's described is simple algebra.

Natural log of constant multiplied by x equals natural log of one plus v squared.

So$$\ln(cx)=\ln(1+v^2).$$

And since v equals y over x.

So$$\ln(cx)=\ln\left(1+\left(\frac{y}{x}\right)^2\right).\qquad(1)$$

That gives us the final function: x squared plus y squared minus c x cubed equals zero.

So$$x^2+y^2-cx^3=0,\qquad(2)$$which comes from exponentiating both sides of (1):$$cx=1+\left(\frac{y}{x}\right)^2,$$multiplying both sides by $x^2$:$$cx^3=x^2+y^2,$$and subtracting $cx^3$ from both sides, which gives (2).


Equation (2) looks like (but is not) the spacetime interval invariant:

$$s^2 = −c^2t^2 + x^2 + y^2 + z^2.$$

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  • $\begingroup$ Yes, Genius makes Einstein sound like he's making things up, with little of his physics explanations making sense except generics, and other physics as clueless - except of Mdm Curie, and Einstein's first wife who, contrary to historical research, maybe made sense of it while completing his papers. It depicted Einstein as a coward, not defending his wife vs his mother. It made him seem to be a happy go lucky guy, totally selfish w/o moral depth. I hope, that from what I know of his later life as a deeply moral man, and his early science as genial, that it was just an unwise youthfulness. $\endgroup$ – Bob Bee Jun 4 '17 at 20:20

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