y = 3e^(2x) - 4sin(2x) Determine whether the function is a solution of the differential equation y^((4)) - 16y = 0

Given,
y=3e^(2x) -4sin(2x)
so,
we have to find
y'=(3e^(2x) -4sin(2x))' =(3e^(2x))'-(4sin(2x))'
=3*2 e^(2x)-2*4 cos(2x)
=6e^(2x)-8cos(2x)
similarly
 
y'' =(6e^(2x)-8cos(2x))'
=6*2 e^(2x)+2*8 sin(2x)
=12 e^(2x)+16 sin(2x)
 
y'''=(12 e^(2x)+16 sin(2x))'
=12*2 e^(2x)+16*2 cos(2x)
=24 e^(2x)+32 cos(2x)
 
y'''' =(24 e^(2x)+32 cos(2x))'
=24*2 e^(2x)-32*2 sin(2x)
=48 e^(2x)-64sin(2x)
 
So lets see whether y'''' -16y=0
=> 48 e^(2x)-64sin(2x) -16(3e^(2x) -4sin(2x))
=48 e^(2x)-64sin(2x) -48 e^(2x)+64sin(2x) =0
so,
y'''' -16y=0