Second-order systems of linear differential equations of the form $$\frac{d^2{\bf x}}{dt^2} = A {\bf x}$$ come up all over the place in physics, economics and engineering. Fortunately, these almost always occur with matrices that are diagonalizable and have real eigenvalues. In this section, we don't have to worry about power vectors or complex eigenvalues! Less fortunately, the solutions to the associated scalar differential equations $$\frac{d^2y}{dt^2} = \lambda y$$ take different forms depending on whether $\lambda$ is positive, negative, or zero. The elliptic case of $\lambda<0$ and the special case $\lambda=0$ are explained in the first video. The hyperbolic case of $\lambda>0$ is explained in the second video.

Once we understand scalar 2nd order differential equations $\frac{d^2y}{dt^2} = \lambda y$, we can understand systems $\frac{d^2{\bf x}}{dt^2} = A {\bf x}$. As with first-order systems, we:

1. Diagonalize A.
2. Define ${\bf y} = [{\bf x}]_\mathcal{B}$.
3. Write the equations in terms of the $y$ coordinates. The result is a bunch of decoupled equations of the form $\frac{d^2 y_j}{dt^2} = \lambda_j y_j$.
4. Solve the decoupled equations, as in the videos above, and
5. Go around the square, converting initial conditions for ${\bf x}$ (namely ${\bf x}(0)$ and $\dot{\bf x}(0)$) into initial conditions for ${\bf y}$ (that is, ${\bf y}(0)$ and $\dot{\bf y}(0)$) into ${\bf y}(t)$ into ${\bf x}(t)$.