Multiplying the differential equation $$\frac{df}{dt} + af(t) = g(t),$$ where $a$ is a constant and $g(t)$ is a smooth function, by $e^{at}$, gives
$$e^{at} \frac{df}{dt} + e^{at} a f(t) = e^{at} g(t),$$
$$\frac{d}{dt} \left( e^{at}f(t) \right) = e^{at} g(t),$$
$$e^{at} f(t) = \int e^{at} g(t) dt,$$
$$f(t) = e^{-at} \int e^{at} g(t) dt.$$

Use this to solve the initial value problem

$\displaystyle \frac{dx}{dt} =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ -2 1 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ 0 4

$x$,

with $x(0) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ 3 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ 5

,

i.e. find first $x_2(t)$ and then $x_1(t)$.

$x_1(t)=$ ,
$x_2(t)=$ .