[tex] \frac{(2N-1)^2}{N(1-N)}=Ce^t [/tex]

The hint says "consider taking a substitution u=N-1/2" ...which is the top bit of the fraction. But what does take a substitution here mean ?

This is a part of a loooong modelling problem which involved an ugly *** integral and gave this equation as the result. I've never evaluated a "quartic equation" so I'm a bit confused about the process.

I have the answer from wolfram so am looking for guidance on how to work it out by hand.