Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means each will be of the form A+ B cos(x)+ C sin(x).

But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we have so that . That equation has general solution . Then so that . Finally, so that just as I said.