Do we expect that "unsolvable" differential equations would have an analytical solution if we simply knew more math? 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.
 A: 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.
