Setting $\displaystyle f(x)=y$ we have...

$\displaystyle y= \sum_{n=1}^{\infty} a_{n}\cdot x^{n}$ (1)

... with the condition...

$\displaystyle y\cdot \ln y = x \rightarrow \ln y = \frac{x}{y}$ (2)

Since is ...

$\displaystyle \frac{d}{dx} \ln \{f(x)\} = \frac {f^{'}(x)}{f(x)}$ (3)

... the condition (2) becomes...

$\displaystyle \frac{d}{dx} \ln y= \frac{y^{'}}{y} = \frac {y - x\cdot y^{'}}{y^{2}}$ (4)

... that can be written as...

$\displaystyle y^{'}\cdot (1+\frac{x}{y}) = \frac{1}{y}$ (5)

The (5) is a first order ODE and we have to find a solution that for (1) has to be analythic... end of the first step

...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$