Calculate the inverse Laplace transform of a right-sided sequence with transfer function

The characteristic equation of G(s) is given by s² − s − 6 = 0, which has two roots at s = 3 and −2. Using the partial fraction expansion, the Laplace transform G(s) is expressed as

$G\left(s\right)=\frac{7s-6}{(s+2)(s-3)}\equiv \frac{k_1}{(s+2)}+\frac{k_2}{(s-3)}\mathrm{.}$

The coefficients of the partial fractions $k_1$k1​ and $k_2$k2​ are given by

$k_1={\left[\right(s+2\left)\frac{(7s-6)}{(s+2)(s-3)}\right]}_{s=-2}={\left[\frac{(7s-6)}{(s-3)}\right]}_{s=2}=4$

and

$k_2={\left[\right(s-3\left)\frac{(7s-6)}{(s+2)(s-3)}\right]}_{s=3}={\left[\frac{(7s-6)}{(s+2)}\right]}_{s=3}=3.$

The partial fraction expansion of the Laplace transform G(s) is therefore given by

$G\left(s\right)=\frac{4}{(s+2)}+\frac{3}{(s-3)}\mathrm{.}$

$g\left(t\right)=(4{\mathrm{e}}^{-2t}+3{\mathrm{e}}^{3t})u\left(t\right)\mathrm{.}$