This week, i have been reading again Boyce and Diprima's textbook on ODEs, and it makes an [informal] statement of what would be a formulation of Abel's Theorem for systems of differential equations. In order to ground the problem, consider the following notation:
\begin{align} \mathbf{x}(t) = (x_{1}(t),\cdots,x_{n}(t)), \text{being the state vector}\\ \dot{\mathbf{x}}(t) = P(t)\mathbf{x}(t), \text{for a matrix }P(t)\text{, and the already defined vector} \end{align}
Consider, also, $\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(n)}$ to be a set of solutions for the equation above, then the wroskian $W[\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(n)}]$ suffices the following differential equation:
\begin{align} \dfrac{dW}{dt} &= tr(P)W \end{align}
Has anyone ever saw a proof of that theorem? Since it involves the derivative of a determinant, I am not sure in how to proceed to the proof.