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 ${\frac {d^{2}y}{dt^{2}}}+2{\frac {xy}{dt}}+3y=f(t)$
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: ${\begin{bmatrix}{\dot {x}}1\\{\dot {x}}2\\\end{bmatrix}}={\begin{bmatrix}0&1\\-3&-2\\\end{bmatrix}}{\begin{bmatrix}x1\\x2\\\end{bmatrix}}+{\begin{bmatrix}0\\1\\\end{bmatrix}}f(t)$
Or, more generally,
$\mathbf {\dot {x}} =\mathbf {Ax+B} f$
This is called the state space representation of the differential equation.