I know how to solve the differential equation $\dot{x} = A x$ when $A$ is a constant $n\times n$ matrix. However, I cannot solve the problem when $A$ also depends on $t$.
To be more specific, $A (t)=\left(\begin{array}{cc} 1 & -1/t\\ 1+t & -1 \end{array}\right)$, where $t>0$.
I can verify that $x(t)=\left(\begin{array}{c} 1\\ t \end{array}\right)$ is a solution. However, in order to find the fundamental solution, I need another linearly independent solution. I tried to set the other solution to be $\left(\begin{array}{c} y_{1}(t)\\ y_{2}(t) \end{array}\right)$ and plugged it into the equation. Then I got the following:
$y_{1}t-y_{2}=\dot{y}_{1}t$ and $y_{1}+y_{1}t-y_{2}=\dot{y}_{2}$. Then, I was stuck.
I also tried $(\det\Phi)^{\prime}=trA\det\Phi$ and got nowhere.