I've read various threads about the logistic map (Verhulst's equation) including:
closed form solutions for the logistic map
https://arxiv.org/pdf/0910.1560.pdf
Logistic Map closed form for $x_{n+1}=rx_n(1-x_n), n\in\mathbb N_0$?
And they all mention that closed form solutions only exist for specific values of r. But what is wrong with the following reasoning:
\begin{gathered} {{x}({n}{+}{1})\hspace{0.33em}{=}\hspace{0.33em}{r}{*}\hspace{0.33em}{x}({n}){(}{1}\hspace{0.33em}{-}{x}({n}){)}}\\ \\but \\{{x}({n}{+}{1})\hspace{0.33em}{=}\hspace{0.33em}{dx}\hspace{0.33em}{+}{x}({n})}\\ \\substituting\hspace{0.33em}gives \\ \\ {{dx}\hspace{0.33em}{+}{x}({n}){=}\hspace{0.33em}{r}{*}\hspace{0.33em}{x}({n}){(}{1}\hspace{0.33em}{-}{x}({n}){)}}\\ \\subtracting\hspace{0.33em}{x}({n})\hspace{0.33em}from\hspace{0.33em}both\hspace{0.33em}sides \\ \\{{dx}\hspace{0.33em}{=}\hspace{0.33em}{r}{*}\hspace{0.33em}{x}({n}){(}{1}\hspace{0.33em}{-}{x}({n}){)}\hspace{0.33em}{-}\hspace{0.33em}{x}({n})}\\ \\we \hspace{0.33em}know\hspace{0.33em}that\\ {{f}{(}{x}{)}\hspace{0.33em}{=}\int{dx}}\\ {{f}{(}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}\int{{r}{*}\hspace{0.33em}{x}{(}{1}\hspace{0.33em}{-}{x}{)}\hspace{0.33em}{-}\hspace{0.33em}{x}}}\\ {{f}{(}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}{-}\frac{1}{6}{x}^{2}\left({{3}{+}{r}\left({{-}{3}{+}{2}{x}}\right)}\right)} \end{gathered}
Clearly, I am missing something HUGE. I know integration is over the set of real numbers whereas Verhulst's equation is intended for integers. But is that a significant difference? If something is valid for the set of real numbers then it should be valid for the set of integers, which is just a subset.