Is there a known solution for the linear equation below? $$ \frac{d\mathbf{y}}{dt} + \mathbf{G}(t) \mathbf{y} = \mathbf{z}(t) $$ The variables $\mathbf{y}$ is a vector of $n$ elements, $\mathbf{z}$ is the external driver term which is also a vector of $n$ elements, and $\mathbf{G}$ is an $n\times n$ matrix that is a function of time $t$.
What I know so far
Assuming $\mathbf{G}(t)$ is diagonalizable, we can decompose $\mathbf{G}=\mathbf{P}\boldsymbol{\Lambda}\mathbf{P}^{-1}$ with diagonal matrix $\boldsymbol{\Lambda}$. If the eigenvector matrix $\mathbf{P}$ can be made constant over time, by defining $\mathbf{u}=\mathbf{P}^{-1}\mathbf{y}$ and $\mathbf{w}=\mathbf{P}^{-1}\mathbf{z}$, we obtain separable linear equations
$$ \frac{d\mathbf{u}}{dt} + \boldsymbol{\Lambda}(t) \mathbf{u} = \mathbf{w}(t) $$
which can be solved using Green's function.
However, if the eigenvector matrix $\mathbf{P}$ can't be made constant, then I'm stuck at the equation below $$ \frac{d\mathbf{u}}{dt} + \left(\mathbf{P}^{-1}\frac{d\mathbf{P}}{dt}(t) + \boldsymbol{\Lambda}\right) \mathbf{u} = \mathbf{w}(t) $$ which, unfortunately, the term besides $\mathbf{u}$ might not be separable (due to the $\frac{d\mathbf{P}}{dt}(t)$ term). I can repeat the exercise by decomposing the matrix besides $\mathbf{u}$, but it seems like there's no guarantee that it will arrive to separable equations.
Is there any way to solve the linear equation above?