A very interesting differential equation.
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as the standard worked example of a function whose derivative is its own inverse, usually as the starting point for asking about other solutions or a more principled method.
Recommended by 1 people across 1 different months, 2020 to 2021.
What it is
Michael Penn works through the differential equation f'(x) = f^(-1)(x), asking when a function's derivative equals its inverse. He guesses a power-law solution f(x) = C x^n and solves for the constants, arriving at an exponent equal to the golden ratio.
What people said
Recommends Karolex · 2020-08 · Mathematics Stack Exchange · on “Is this a good intuition for the general approach to solving non-trivial differential equations?”
Recently I have watched the video "A very interesting differential equation." by Michael Penn (link: [this video] ) in which he solves he following differential equation: $$f'(x)=f^{-1}(x)$$ The way he approaches solving this differential equation is by assuming that the solution $f(x)$ must be in the same "class" of functions as $f'(x)$ and $f^{-1}(x)$ . The function that satisfies this criteria is $f(x)=Cx^n$ . The exact solution to this differential equation is $$f(x)=\sqrt[\huge{\phi}]{\frac{1}{\phi}}\cdot x^{\Large{\phi}}$$ where $\phi$ is the golden ratio. I have a very basic understanding of abstract algebra and I am trying to get a good intuition for why the assumption that $f(x)$ has to
Cites Mrcrg · 2020-06 · Mathematics Stack Exchange · on “A Function $f$ such that $f' = f^{-1}$”
Example of a function that its derivative is the same as its inverse. That is $$f'(x)=f^{-1}(x)$$ In this video [this video] Michael Penn, shows a technique and presents a function with that property. The example is $$f(x)=\sqrt[\varphi]{\dfrac{1}{\varphi}} \cdot x^{\varphi}$$ where $\varphi=\dfrac{1+\sqrt{5}}{2}$ But his technique is based on an intelligent guess , is it possible to solve this problem in another way and find another function with this property?
Cites dimpap5555 · 2020-06 · Mathematics Stack Exchange · on “Solve Differential Equation $ f^{-1}(x)=f'(x) $”
$$ f^{-1}(x)=f'(x)$$ Where $f^{-1}:$ Inverse of f I know for a fact there is a specific solution that is demonstrated on this video [this video] and specifically: $$ f(x) = x^{φ}\sqrt[φ]{\frac{1}{φ}} $$ Where $φ=\frac{1+\sqrt{5}}{2}$ is the golden ration. What I wish to do, is to find (if they exist) the rest of the solutions to the equation. In case there is no other, I want to at least prove that there is no other solution. For the record, I seek a conventional solution with a mathematical proof backing it, and not just through making "educated" guesses.
Cites russian bot · 2020-12 · Mathematics Stack Exchange · on “When is the inverse function of the derivative equal to the integral?”
Note on duplicates This [LINK] question is not a duplicate; it asks when the inverse of the derivative equals the function , mine asks when the inverse of the derivative equals the integral . The question Let $f$ be any function. Define $F(x)$ to be the integral of $f(x)$ , and $(f')^{-1}$ to be the inverse function of the derivative of $f$ . When is $$F(x)=(f')^{-1}?$$ I don't know much about how to solve this equation. Any help would be appreciated. $(f')^{-1}$ is not the reciprocal of $f'$ . I could do this, but I don't know where to begin from. Any hint of where to start from would also be helpful. My thoughts and work Does such a function even exist? Why/why not? From the initial equati
Cites NoiX07 · 2021-02 · Mathematics Stack Exchange · on “Derivative and inverse”
I had to mention this à-propos [this video] whose solution is of the form suggested by Chritian Blatter!! Unfortunately there is no video about our problem...\\ By the way, one could also write the equation with both $\langle -1 \rangle$ on the l.h.s., it looks better...