Find functions which are fulfilling:
for all .
Prove that if are fulfilling the above equation then determined by . How?
This is how I might attempt to do this problem. I will assume that both and are smooth. Taking the natural log of both sides gives
Call the term on the RHS . Differentiating both side wrt and gives
Thus, where and are constant.
Now set and this gets you the form for Then substitute into the original functional equation to determine .