Consider the second order differential equation with initial conditions $$u''+2.5u' + 2.5u = 8\sin(3t), \qquad u(1)=-4.5, \qquad u'(1)=-3 .$$ Without solving it, rewrite the differential equation as an equivalent set of first order equations. In your answer use the single letter u to represent the function $u$ and the single letter v to represent the "velocity function" $u'$. Do not use $u(t)$ or $v(t)$ to represent these functions. Expressions like $\sin(t)$ that represent other functions are OK.

$u' =$
$v' =$

Now write the first order system using matrices:

$\displaystyle \frac{d}{dt}$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $u$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $v$

$=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $u$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $v$

$+$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

.

The initial value of the vector valued solution for this system is:

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $u(1)$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $v(1)$

$=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

.