Question

Find the complete time-domain solution y(t) for the rational algebraic output transform Y(s):_________

Answer 1

Answer:

y(t)= 11/3 e^(-t) - 5/2 e^(-2t) -1/6 e^(-4t)

Step-by-step explanation:

$Y(s)=\frac{s+3}{(s^2+3s+2)(s+4)} + \frac{s+3}{s^2+3s+2} +\frac{1}{s^2+3s+2}$

We know that $s^2+3s+2=(s+1)(s+2)$, so we have

$Y(s)=\frac{s+3+(s+3)(s+4)+s+4}{(s+1)(s+2)(s+4)}$

By using the method of partial fraction we have:

$Y(s)=\frac{11}{3(s+1)} - \frac{5}{2(s+2)} -\frac{1}{6(s+4)}$

Now we have:

$y(t)=L^{-1}[Y(s)](t)$

Using linearity of inverse transform we get:

$y(t)=L^{-1}[\frac{11}{3(s+1)}](t) -L^{-1}[\frac{5}{2(s+2)}](t) -L^{-1}[\frac{1}{6(s+4)}](t)$

Using the inverse transforms

$L^{-1}[c\frac{1}{s-a}]=ce^{at}$

we have:

$y(t)=11/3 e^{-t} - 5/2 e^{-2t} -1/6 e^(-4t)$