13 Matrix form of the damped harmonic oscillator

We introduce a new state variable $v$, the velocity, so $y^{\prime }=v$. Then we have a pair of equations $y^{\prime }=v$ and $v^{\prime }=-\beta v-\gamma y+u$. We put these together in a matrix equation

$$X=\left[\begin{array}{c}\hfill y\hfill \\ \hfill v\hfill \end{array}\right],U=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill u\hfill \end{array}\right]$$

so the differential equation is

$$\frac{dX}{dt}=AX+U.$$