State Space Form

In my Signals and Systems II class at University of Idaho, we are learning about the state space form of representing a solution to a LTI differential equation. I'll add more here soon. Consider the differential equation $\frac{d^{2}y}{d{t}^2}+2\frac{xy}{dt}+3y=f(t)$
or
$\ddot{y}+2\dot{y}+3y=f(t)$
The state of this equation can be described using what is called state space form. State space form gives the blah blah more here.

let $x1=y$
let $x2=\dot{y}$
so $\dot{x}2=\ddot{y}$

We can now re-write the equation above to be:
$\dot{x}2+2x2+3x1=f(t)$
so
$\dot{x}2=-3x1-2x2+f(t)$
and from the definition above
$\dot{x}1=x2$

We can take this and put it into matrix form: $\left[\begin{array}{c}\dot{x}1\\ \dot{x}2\\ \end{array}\right]=\left[\begin{array}{cc}0& 1\\ -3& -2\\ \end{array}\right]\left[\begin{array}{c}x1\\ x2\\ \end{array}\right]+\left[\begin{array}{c}0\\ 1\\ \end{array}\right]f(t)$
Or, more generally,
$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}\mathbf{+}\mathbf{B}f$
This is called the state space representation of the differential equation.