A Differential Equation in Terms of y

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[larzcaetano]

Ahhh, Cardano at the differential equation's heart. That is great. I don't like the approach of doing separation of variables nor the approach of flipping it. I usually like to go for something a little more rigorous. Although doing separable solutions or finding the reciprocal works, it's fun to follow the rigour. Amazing job!!! Funny how the (a + b)^3 Cardano came up with shows up in the end :)

[mokouf]

Nice job flipping a dy/dx = f(y) into dx/dy = 1/f(y)
This way is easy even for those who just started learning integrals (provided they remember the identities)

[Roq-stone]

I’m glad you cleared up that the dx/dy is not a division.

[allanmarder]

If you have a cubic of the form y^3 + p*y + q = 0 the solution is y= cube root (-q/2 + sqrt[ (q/2)^2 + (p/3)^3 ]) + cube root (-q/2 - sqrt[ (q/2)^2 + (p/3)^3 ]).
From the video we have y^3 -3y - (3x+K) = 0. Plugging these values into the above we have:
y= cube root ((3x+K)/2 + sqrt[ ((3x+K)/2)^2 - 1 ]) + cube root ((3x+K)/2 - sqrt[ ((3x+K)/2)^2 - 1 ])

[broytingaravsol]

very comprehensive, the form of separable ones

[drneuropharmacology]

Hello, what software do you use to write your equations?

[vladimirkaplun]

Niccolò Tartaglia sends you greetings. Along with Gerolamo Cardano and Ludovico Ferrari.

[scottleung]

Well, it's good we were asked to solve for x in terms of y and not vice versa!