I generally understand how to solve linear systems using x' = Ax where x is a vector, and A is a matrix. However, I am lost when the A matrix becomes periodic. I am going to give an example, and show what my professor explained was the correct answer, but there are certain gaps in my knowledge.
Given the equation
x1' = (1+cos(t)/(2+sin(t)))x1
x2' = 2x1 - x2
I understand that the A(t) matrix would be
$\begin{bmatrix}(1+cos(t)/(2+sin(t))) & 0\\2 & -1\end{bmatrix}$
Making it so that technically the eigen values would be (1+cos(t)/(2+sin(t))) and -1
However, my professor then goes on to explain what x1 and x2 is and I am totally lost on how to get there.
x1 = c1et(2+sin(t))
x2 = c1et[(2/5)(2sin(t)-cos(t)+2] + c2e-t
Which would then lead to the X matrix where X is defined as
$\begin{bmatrix}e^t(2+sin(t)) & 0\\e^t[(2/5)(2sin(t)-cos(t)+2] & e^{-t}\end{bmatrix}$
I was thinking that when x1 = 0, that x2' + x2 = 0, therefore x2 = e-t which would make part of the X matrix make sense. However, I am completely lost on how x1 and x2 are found. If someone could explain to me how Floquet is used to find the matrix X that would be so helpful. Thank you.