Question

Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your answer as a function of t.)

ℒ⁻¹{(s/(s² + 3s − 4)}

Answer 1

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

$=\frac{A}{s-a}+\frac{B}{s+4}=\frac{(A+B)s+4A-B}{(s-1)(s+4)}$

we have a system:

$\begin{cases} A+B=1\\ 4A-B=0 \end{cases}$

$5A=1$

$A=\frac{1}{5}$

$B=\frac{4}{5}$

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

$L^{-1}(\frac{s}{s^2+3s-4})=L^{-1}(\frac{1}{5(s-1)})+L^{-1}(\frac{4}{5(s+4)})=$

$=\frac{1}{5}e^t+\frac{4}{5}e^{-4t}$