For an ordinary differential equation$$f(y^{(n)}(x), y^{(n-1)}(x),...,y''(x),y'(x),y(x),x) = 0$$where $f$ can be written in a closed form using only elementary functions, is it possible for its solution $y(t)$ to grow as fast as tetration or faster?
By "written in a closed form using only elementary functions", I mean that there only finite composition of addition, multiplication, exponentiation, exponential and trigonometric functions and their inverses are used, i.e. no tricks like infinite sums.
By "growing as fast as tetration or faster", I mean that there is $k\in\mathbb N$ and $a >1$, such that$y(x) \ge a \uparrow\uparrow \lfloor x\rfloor$ for all $x \ge k$.
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