Coupled differential equations

Coupled differential equations

The question I have trouble with is from Intro to electrodynamics by
Griffiths
http://imgur.com/a/sfBlt
I do not understand how the author obtained the general solution
$y(t) = C_1 \cos (\omega t) + C_2 \sin (\omega t) + (E/B)t + C_3$
$z(t) = C_2 \cos (\omega t) - C_1 \sin (\omega t) + C_4$,
and then found that
$y(t) = \frac{E}{\omega B}(\omega t - \sin(\omega t))$, and the
corresponding equation for $z(t)$, since plugging in $y(0) = z(0) = 0$
yielded $C_1 = C_2 = C_3 = C_4 = 0$ for me.
Thanks in advance.