How to solve transfer function of a bandpass filter with Butterworth approximation?
Volki, be aware that the design will be rather involved.
Here is a short summary of the main steps (active filter design):


Based on the given values (center frequency, bandwidth, midband gain, attenuation values) you must use existing formulas for the bandpass-lowpass transformation. As a result, you have corresponding lowpass requirements and you can calculate the corresponding lowpass order (n=2,3,..).
Applying again corresponding equations you can transform the lowpss poles back to the bandpass poles.
Now you have two alternatives for pole allocation (depending on the required filter order):


Series connection of active lowpass and highpass sections
Active bandpass stage (or series connection of several stages).  





For passive topologies, you must rely on filter catalogues. In any case - I strongly recommend to make use from filter design programs which are available. For example:

Passive: AADE;
Active: Filter Free, FilterLab, FilterPro