Given $$ \dot{y} = A(t)y,\quad A(t)= \begin{pmatrix} 1 + \frac{\cos t}{2 + \sin t} & 0 \\ 1 & -1 \end{pmatrix}, \quad y\in\mathbb{R}^2, $$
I am supposed to calculate the flow $\Phi^{t,0}$, then calculate the monodromy operator $M$ as $M=\Phi^{2\pi,0}$, and then confirm that $$ M=e^{2\pi\Omega},\quad\Omega=\begin{pmatrix}1 & 0\\\frac{4}{5} & -1\end{pmatrix}.$$
I did all this, but my calculation of $M$ is different from what I'm supposed to confirm.
The steps/assumptions I made:
- $\Phi^{t,0}=\Psi(t)[\Psi(0)]^{-1}$, where $\Psi(t)$ is the fundamental matrix of $A(t)$.
- To get the fundamental matrix, I first solved the first equation of the system, which is $\frac{dy_1}{dt}=A_{11}\cdot y_1$ using separation of variables, and then the second equation $\frac{dy_2}{dt}=A_{21}y_1-A_{22}y_2$, inwhich I could fill in $y_1$, using Lagrange's variation of constants. I now got expressions for $y_1$ and $y_2$ in terms of $y_{1,0}$ and $y_{2,0}$ (to be clear: $y_{1,0}=y_1({t_0})$):
$$y_1=y_{1,0}\frac{1}{2}e^t(\sin(t)+2)),\quad y_2=y_{2,0}e^{-t} + y_{1,0}\frac{1}{2}(e^{t} - \frac{1}{5} e^{t} (\cos t - 2 \sin t) - \frac{4}{5}e^{-t})$$
- So I then figured that the fundamental matrix, which consists of linearly independen solutions, must be
$$\Psi(t)= \begin{pmatrix} \frac{1}{2}e^t(\sin(t)+2) & 0 \\ \frac{1}{2}(e^{t} - \frac{1}{5} e^{t} (\cos t - 2 \sin t) - \frac{4}{5}e^{-t}) & e^{-t} \end{pmatrix}. $$
(I have confirmed that these two columns are indeed solutions to the system at the start of this question. But are they also linearly independent?)
Now, assuming that I didn't make any miscalculations, and these steps make sense for calculating the flow, I get that
$$ M=\Phi^{2\pi,0}=\Psi(2\pi)[ \Psi(0)]^{-1}=\Psi(2\pi)[I]^{-1}=\Psi(2\pi). $$
From this we see that $M_{12}=0$, while the topright element of $e^{2\pi\Omega}$ does not equal zero (according to Matlab Online), so I must have made a mistake in my calculations, methods, or both. If I need to calculate the fundamental matrix in another way, or must not use the fundamental matrix to calculate the flow, please advise me on what to do.