Determine the inverse laplace transforms of
s+7/s^2+4s+3
and
3s-8/s^2+4
1: Complete the square down below and you can split it into two LTs of the form:
$\displaystyle \frac {s-b}{(s-b)^2 + a^2}, \frac {1}{(s-b)^2 + a^2}$
multiplied by whatever constants. Or they could be - a^2 on the bottom. Whatever, you use your inverse tables and get $\displaystyle e^{bt} \sin a t / a$ or it could be cos or sinh or cosh, whatever, see what you get when you rip it apart.
You can simply factor:
1. (s+7)/[s^2+4s+3] = (s+7)/[(s+3)(s+1)]
= (s+3)/[[(s+3)(s+1)] + 4/[(s+3)(s+1)]
= 1/(s+1) + 4/[(s+3)(s+1)]
y(t) = e^(-t ) + 4 (e^(-3t)-e^(-t))/-2
y(t) = 3e^(-t) - 2e^(-3t)
2. write as 3s/(s^2+4) - 8 /(s^2+4) = 3{s/(s^2+4)} - 4{2 /(s^2+4)}
y(t) = 3cos(2t) - 4sin(2t)