Section 5.2 is about systems of first-order linear differential equations, of the form $$\frac{d{\bf x}}{dt} = A {\bf x}.$$ These are closely related to the scalar differential equations $$\frac{dy_i}{dt} = \lambda_i y_i,$$ whose solution is $y_i(t) = y_i(0) e^{\lambda_i t}$. If this isn't clear to you, then you should watch a re-run of the following video from Chapter 1, which reviews this differential equation. (The first half of the video talks about the general strategy of decoupling. We start solving the differential equation around the 3 minute mark.)

In the second video, we see how to solve the system of coupled equations when $A$ is diagonalizable and has real eigenvalues. The general solution is $${\bf x}(t) = \sum_{i=1}^n c_i e^{\lambda _i t} {\bf b}_i,$$ where the $c_i$'s are arbitrary constants and the ${\bf b}_i$'s are eigenvectors with eigenvalue $\lambda_i$.

When $A$ has complex eigenvalues, say $\lambda=a+bi$, then the procedure is the same, but we have to remember that $$e^{(a+bi)t} = e^{at} e^{bit} = e^{at}(\cos(bt) + i \sin(bt)).$$ Our solutions will involve both exponential growth and oscillation. The real part of the eigenvalue gives the growth rate of a term, and the imaginary part gives the frequency of oscillation.

As with difference equations, there is a simple closed-form solution to $d{\bf x}/dt = A {\bf x}$, namely ${\bf x}(t) = e^{At} {\bf x}(0)$. The tricky thing is computing $e^{At}$. If $A$ is diagonalizable, then $$ e^{At} = P e^{Dt} P^{-1}.$$ The three factors correspond exactly to the three conversions $${\bf x}(0) \, {\mathop{\Longrightarrow}^{P^{-1}}} \, {\bf y}(0) \, {\mathop{\Longrightarrow}^{e^{Dt}}} \, {\bf y}(t) \, {\mathop{\Longrightarrow}^P} \, {\bf x}(t).$$

When $A$ isn't diagonalization, we find the matrix of $e^{At}$ in the $\mathcal{B}$ basis by studying how $e^{At}$ acts on eigenvectors and on power vectors. Just as with $A^n$, this was explained in the third video for Section 4.9.

The last video for this section works an example with $A$ non-diagonalizable.