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In my engineering studies and while reading the book Chaos, I see a lot of mentions of complicated differential equations without solutions.

For example, the equation $$\frac{dx}{dt}+\sin(x(t))=\sin(wt)$$ does not have an analytical solution as far as I know.

Is there hope that if we had more functions at our disposal (for example, more functions like sine, hyperbolic sine, etc.) we would be able to find such a solution? Or is something like this fundamentally unsolvable for some reason?

If it would be possible, are mathematicians working to discover these new mathematical terms? It fascinates me that we don't have the math to cleanly describe the three-body problem, for example, and it's hard to imagine that a clean solution wouldn't exist if we simply knew more.

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jgholder
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    Well, we could always create a function that's defined as the solution to that equation. A lot of special functions are defined this way. – eyeballfrog Aug 16 '21 at 01:31
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    Non-elementary functions that turn up "often enough" as, for instance, solutions to differential equations get particular attention and are identified in various "families" of special functions. But there are all manner of transcendental functions about which we know little: there is a vast amount to "explore" and only so many "explorers". How many of those other functions are really worth the effort of characterizing more fully is itself part of what we are still ignorant of... –  Aug 16 '21 at 01:34
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    The "problem" with referring to "chaotic" (complex) systems is that there isn't a function which describes the behavior of the system, but rather a "trajectory" for each possible initial condition. (What makes the behavior "irreproducible" is that it is not physically possible to know exactly what the initial conditions are, but the trajectory is dependent upon them.) The system of DEs is the description of the behavior, but the "outcome functions" can be innumerable. –  Aug 16 '21 at 01:43
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    Are mathematicians working to discover these new functions? Not really, at least not specifically. It is a rather dull form of "discovery", study a function defined by a given equation and give it a name. The usefulness of it is also limited, equations with similar behavior can have wildly different closed form solutions, and those can be too cumbersome to be of any use. If there is a point to it it is to identify functions that pop up in multiple unexpected places, see Painlevé transcendents, not those for some particular equation. – Conifold Aug 16 '21 at 03:46
  • Maybe this, this, this, this and this will help (last two are more advanced answers to similar questions). – Matthew Cassell Aug 16 '21 at 06:43

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Check out this

Differential Equations without Analytical Solutions

When people say no analytic solution, they usually don't mean that they haven't found an analytic solution yet. They mean they've proven any analytic function will not satisfy the ODE. There aren't more analytic functions out there we don't know of. Even in some specially cooked up examples (like in the link), you can find ODEs with pretty simple closed-form solutions that are not analytic

An interesting find on the 3-body problem wiki page: https://en.wikipedia.org/wiki/Three-body_problem#cite_note-13

"There is no general closed-form solution to the three-body problem,[1] meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations."

"However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a power series in terms of powers of $t^{\frac{1}{3}}$.[13]"

An cool example to show even if there's no analytic solution in the traditional sense, with cleverness you can find a solution that's kinda like an analytic solution.

Brian Lai
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  • i feel like you are conflating closed-form/“analytical” and analytic ie locally expressible as a power series. The first link is talking of analytical functions, but later you talk about a power series in fractional powers of $t$ – Calvin Khor Aug 16 '21 at 01:44
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    When people say "analytic solution" in the OP context they mean "closed form solution", not locally holomorphic one. "Closed form" most often means in elementary functions, but can also include known special functions, so if extra special functions are added what didn't have closed form solution can acquire it. – Conifold Aug 16 '21 at 03:34