Answer
See explanation
Work Step by Step
$(a)$ If $y = sin 2x$ then $y' = 2 cos 2x$ and $y'' = -4 sin 2x$
Substitute into the equation $y'' + 4y = 0$
$ -4 sin 2x + 4 sin 2x = 0$
$ 0 = 0 $ Verified
If $y = cos 2x$ then $y' = -2 sin 2x$ and $y'' = -4 cos 2x$
Substitute into the equation $y'' + 4y = 0$
$- 4 cos 2x + 4 cos 2x = 0$
$ 0 = 0$ Verified
$(b)$ If $y = c_1 sin 2x + c_2 cos 2x$
then $y' = 2c_1 cos 2x - 2c_2 sin 2x$
and $y'' = -4c_1 sin 2x - 4c_2 cos 2x$
Substitute into the equation $y'' + 4y = 0$
$(-4c_1 sin 2x - 4c_2 cos 2x) + 4(c_1 sin 2x + c_2 cos 2x) = 0$
$-4c_1 sin 2x - 4c_2 cos 2x + 4c_1 sin 2x + 4c_2 cos 2x = 0$
$ 0 = 0 $ Verified
