I am trying to find a general solution (if it exists) to the system of equations:
$$(fg)'=f'g'$$
$$(fg)''=f''g''$$
we can obviously rewrite these as:
$$fg'+gf'=f'g'$$
$$fg''+gf''+2f'g'=f''g''$$
I am not really sure how to go from here without substituting one into the other and seeing what happens e.g:
$$fg''+gf''+2fg'+2gf'-f''g''=0$$

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It is also easy to assume certain things about each function and get a specific solution out e.g. $g''=0$ but this does not help with the general solution. My final thought was to write it as:
$$\frac{g''}{g'}=\frac{f''}{f''-f'}$$