Find the fundamental matrix of the system of odes:
$$\left[ \begin{array}{c} x \\ y\\ \end{array} \right]'= \left[ \begin{array}{ccc} -1+\frac{3}{2}\cos^2t & 1-\frac{3}{2}\sin t\cos t \\ -1-\frac{3}{2}\sin t \cos t & -1+\frac{3}{2}\sin^2t\\ \end{array} \right]\left[ \begin{array}{c} x \\ y\\ \end{array} \right].$$
Attempt. The coefficient matrix is not constant, so we may not use the eigenvalue-eigenvector method. I wrote the system as:
$$x'= \left(-1+\frac{3}{2}\cos^2t\right)x+\left(1-\frac{3}{2}\sin t\cos t\right)y$$ $$y'=\left(-1-\frac{3}{2}\sin t \cos t\right)x+\left(-1+\frac{3}{2}\sin^2t\right) y$$
and since both have non-zero coefficients, maybe a combination of these two equations would lead us to a solvable form. I didn't end up to such a combination though. Another idea was to differentiate the first one and reach a second order linear ode for $x$ (but also led me to non-constant coefficients).
Thanks for the help.