Answer
See explanation
Work Step by Step
$(a)$ If $y = e^{-2x} sin 3x$
$y' = e^{-2x}(-2 sin 3x+ 3 cos 3x)$
$y'' = e^{2x}(-5 sin 3x-12 cos 3x)$
substitute into the equation $y'' + 4y' + 13y = 0 $
$e^{2x}(-5 sin 3x-12 cos 3x) + 4e^{-2x}(-2 sin 3x+ 3 cos 3x) +13e^{-2x} sin 3x = 0 $
$0 = 0$ verified
If $y = e^{2x} cos 3x $
$y' = e^{-2x}(-3 sin 3x - 2 cos 3x) $
$y'' = e^{-2x}(12 sin 3x - 5 cos 3x)$
substitute into the equation $y'' + 4y' + 13y = 0 $
$e^{-2x}(12 sin 3x - 5 cos 3x) + 4e^{-2x}(-3 sin 3x - 2 cos 3x) + 13e^{-2x} cos 3x = 0$
$0 = 0$ verified
$(b)$ If $y = e^{-2x}(c_1 sin 3x+c_2 cos 3x) $
then $y' = e^{-2x}[-(2c_1+3c_2) sin 3x+(3_c1-2c_2) cos 3x] $
and $y'' = e^{-2x}[(-5c_1+ 12c_2) sin 3x - (12c_1+5c_2) cos 3x ] $
substitute into the equation $y'' + 4y' + 13y = 0 $
$e^{-2x}[(-5c_1+ 12c_2) sin 3x - (12c_1+5c_2) cos 3x ] + 4 e^{-2x}[-(2c_1+3c_2) sin 3x+(3c_1-2c_2) cos 3x] + 13 e^{-2x}(c_1 sin 3x+c_2 cos 3x) = 0$
$ 0 = 0$ verified
