Answer
The function is a solution of the differential equation
Work Step by Step
Start by finding the fourth derivative of y.
$y=3e^{2x} - 4\sin2x$
$y^{'} = 6e^{2x} +8\cos2x $
$y^{''} = 12e^{2x} +16\sin2x$
$y^{(3)} = 24e^{2x} -32\cos2x$
$y^{(4)} = 48e^{2x} -64 \sin2x$
Substitute back into the differential equation
$y^{(4)} - 16y =0$
$(48e^{2x} -64 \sin2x)-(48e^{2x} +64 \sin2x) = 0$
$0=0$, The function is a solution of the differential equation.
