Chapter 8.1, Problem 21P

Determine which of the following sets of vectors is a basis for the solution space to the differential equation $y^{″}-16y=0$:

$S_1=\left\{e^{4x}\right\}$, $S_2=\left\{e^{2x},e^{4x},e^{-4x}\right\}$, $S_3=\left\{e^{4x},e^{2x}\right\}$, $S_4=\left\{e^{4x},e^{-4x}\right\}$, $S_5=\left\{e^{4x},7e^{4x}\right\}$, $S_6=\left\{\cosh 4x,\sinh 4x\right\}$.