In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
2026-03-25 11:16:54.1774437414
Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$
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$$f'(x)=\frac{f(x)}{x (f(x)+1)} \iff \frac{f'(x) (f(x)+1)}{f(x)}=\frac{1}{x}\stackrel{\int}{\iff}\log f(x)+f(x)=\log x+c_1$$
Solve for $f(x)$.