Topic: Signals and Systems
Inverse of Laplace transform of $\frac{10}{s^{2}+5s+4}$ is
- A. $10[e^{-t}+e^{-4t}]$
- B. $10[e^{-t}-e^{-4t}]$
- C. $\frac{10}{3}[e^{-t}+e^{-4t}]$
- D. $\frac{10}{3}[e^{-t}-e^{-4t}]$
Answer: D. $\frac{10}{3}[e^{-t}-e^{-4t}]$
Explanation: Factor the denominator: $s^2 + 5s + 4 = (s+1)(s+4)$.
Perform partial fraction decomposition: $\frac{10}{(s+1)(s+4)} = \frac{A}{s+1} + \frac{B}{s+4}$.
$A = \left. \frac{10}{s+4} \right|_{s=-1} = \frac{10}{3}$
$B = \left. \frac{10}{s+1} \right|_{s=-4} = \frac{10}{-3} = -\frac{10}{3}$
$F(s) = \frac{10/3}{s+1} - \frac{10/3}{s+4}$.
Taking the inverse Laplace transform: $f(t) = \frac{10}{3}[e^{-t} - e^{-4t}]u(t)$.