Chapter 1 - Introduction - 1.3 Classification of Differential Equations - Problems - Page 25: 16

$r = +1 or -1$.

We must find $r$ such that the differential equation $y''-y = 0 $ has solutions of the form $y = e^{rt}$. Knowing that, in general, $(d/dt) e^{rt} = re^{rt}$, (and so $(d^{2}/dt^{2}) e^{rt} = r^{2}e^{rt})$, by simple substitution, the given differential equation becomes $r^{2} e^{rt} = e^{rt}$, which is satisfied by $r = +1 or -1$.