I'm a college sophomore taking differential equations and I'm having trouble understanding the notation in the part of the course involving systems of differential equations.
All the notation for the notes seem to involve these $x_n$ terms:
$\mathbf{X} = \begin{pmatrix} x_1\\ x_2\\ ...\\ x_n \end{pmatrix}$
What do the $x_1, x_2, ...,x_n$ represent? I'm used to single differential equations that have dependent variables $x$ and/or $y$ as functions of independent variable $t$, but now they're nowhere to be found and we're only using shorthand notations like $\mathrm{X' = AX}$.
I don't understand how they related to the single equations we were dealing with earlier in the course.
The $x_1,\ldots,x_n$ are the $n$ functions in your system of linear equations $X'=AX$. This last expression is simply a (very useful) short hand for \begin{align} x_1'&=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ \ \\ x_2'&=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ \ \\ &\vdots\\ \ \\ x_n'&=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \ \\ \end{align}