Find a solution to the initial value problem

$$y^{\,\prime} + \sin(t) y = g(t), \ \ \ y(0) = 4,$$
that is continuous on the interval $\lbrack 0, 2\pi \rbrack$ where

$$g(t) = \left\lbrace \begin{array}{rcl} \sin(t) && \mathrm{if} \ 0 \leq t \leq \pi, \\ -\sin(t) && \mathrm{if} \ \pi < t \leq 2\pi. \end{array} \right.$$

$y(t) =$ $\displaystyle \Bigg\lbrace$

$\mbox{ if } 0 \leq t \leq \pi,$  $\mbox{ if } \pi < t \leq 2\pi.$