constant evaluation when using differential equations. This is regards to constant evaluation when using differential equations.


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*A solution is given to be:
$$y=(e^{2x}+e^x )  \ln⁡(1+e^{-x} )-(c_1+1) e^x+(c_2-1) e^{2x}$$

*A simplified solution in an answer book is given as:
$$y=(e^{2x}+e^x )  \ln⁡(1+e^{-x} )+(c_1 ) e^x+(c_2 ) e^{2x}$$
There is a change in sign of $c_1$ in the third term. $C_1$ is a constant and not specified to be positive or negative or is it supposed to be positive and that information is simply not specified. I never know how to interpret this kind of results. Can someone explain, please? Thank you.
Sincerely,
Mary A. Marion

Both solutions are correct and they are equivalent. The constants $C_1$ and $C_2$ are just place holders for numbers to be found from initial values and you may as well call them $-C_1-1$ or $C_2+1$
Once the initial values are given the constants are found and the final result is unique regardless of the notations for constants. 
