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Recently I have watched the video "A very interesting differential equation." by Michael Penn (link: https://www.youtube.com/watch?v=rNUfiQgj6ZI&t=308s) 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 be in the same "class" as $f'(x)$ and $f^{-1}(x)$ leads to the correct solution. My current attempt at explaining this approach is the following (albeit with terrible mathematical terminology):

The reason why we're looking for a function where $f'(x)$ and $f^{-1}(x)$ belong to the same "class" of functions is that then both the derivative and inverse of that function share the same algebraic structure (or similar algebraic properties), which makes the equation solvable.

If so, is it then possible to prove the existence of solutions to such equations by analyzing the "algebraic behavior" or properties of the derivation operator and the algebraic properties of all known classes of functions? Is this for example (in a very simplified way) how we prove that $F(x)=\int \exp(-x^2) dx$ cannot be written with our known functions?

I am not looking for a full mathematical explanation, though I would really appreciate if someone can link me a related question, wiki links or literature that explain the solving of differential equations intuitively.

Susp1cious
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  • It's just a guess, it happened to work in this case but it would be easy to find examples where it doesn't. – Qiaochu Yuan Aug 30 '20 at 21:10
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    I don't think this answers your broader questions, but in the problem you mention, $f'$ and $f^{-1}$ are literally equal, so they must be in the same class as eachother for a solution. And at least on an intuitive level for solving differential equations, you would want a class to be closed under differentiation. So if you have a class of functions that contains their own inverses, that might be a good place to look. – Mark S. Aug 30 '20 at 21:11

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Sometimes when we guess a solution to a differential equation, we get it right. Take for instance the differential equation $$x^2y''(x)+2x~ y'(x)+k^2x^2y=0$$ (which arises when solving the wave equation in spherical coordinates) Although I have no idea how to actually construct the solution (despite taking a course on differential equations, in particular second order linear ones), trying the guess solution $$y(x)=\frac{e^{ax}}{x}$$ will produce a correct result. Sometimes we're able to make good guesses, other times not.

And with regard to showing $\int\exp(-x^2)\mathrm{d}x$ has no closed form solution, this has already been discussed at length on this site.

K.defaoite
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