I'm trying to find the fundamental matrix for the following ODE system:
$$
y'=\left(
\begin{matrix}
3-\sin^2 t&\cos t\\
\sin t+1&\cos t\sin t
\end{matrix}
\right)y.
$$
If one can come up with one solution, then the Liouville's formula can be used to find another one. But I have no idea how to find one.
Is there a general method for solving $x'=A(t)x$ when $A(t)$ is nonconstant periodic?
[Added:] Since the coefficient matrix is periodic, one might want to restrict the question a little bit. I'm wondering if one can transform it to the Hill differential equation, which can give a solution.
Best Answer
The Magnus expansion is one approach for this in general.
wikipedia
It is probably not the kind of closed form that you would want. The reason for the complexity is that the matrix at one point in time might not commute with the matrix at a different point in time. So you get a huge ugly series expansion in terms of integrals of nested commutators. This has theoretical interest, but is not of much use for numerical solutions, nor for really understanding what is happening.