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I recognize the differential function equation you are discussing because it is produced by power towers of height x.

Let $f(x)=a^x$, then $f^n(x)$ produces a power tower.

Typically functional equations are solved by having some preexisting idea as to the form of the solution. I do recommend "Iterative Functional Equations" by Kuczma if you haven't read it.

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I recognize the differential function equation you are discussing because it is produced by power towers of height x. Typically functional equations are solved by having some preexisting idea as to the form of the solution. I do recommend "Iterative Functional Equations" by Kuczma if you haven't read it.