I'm first to find the units of all model parameters and subsequently non-dimensionalise the equation. We're given that $y$ and $t$ have units of time whilst $c$ has units of length (with $c$ taken to be a model parameter and $e^x=exp(x)$). The equation is:
$$ y + b{e^{\gamma t}}=c $$
I understand that to begin with I should re-write the two variables $y$ and $t$ as, say:$$y=y^*y'$$$$t=Tt'$$ Beyond this I don't really know what I should do, any help would be appreciated.
You can't have $y$ a time and $c$ a length unless there is a multiplicative constant to fix that. The units on all terms that are added or subtracted need to be the same.