Question

L-1{(e-2s)/[s²(s-1)]}

Answer 1

$$\dfrac{1}{s^2(s-1)}=\dfrac{A}{s}+\dfrac{B}{s^2}+\dfrac{C}{s-1}$$

$$=\dfrac{As(s-1)+B(s-1)+Cs^2}{s^2(s-1)}$$

$$=\dfrac{As^2-As+Bs-B+Cs^2}{s^2(s-1)}$$

$$s^2:A+C=0$$

$$s^1:-A+B=0$$

$$s^0:-B=1$$

$$A=-1, B=-1, C=1$$

$$L^{-1}(\dfrac{e^{-2s}}{s^2(s-1)})=H(t-2)L^{-1}(-\dfrac{1}{s}-\dfrac{1}{s^2}+\dfrac{1}{s-1})(t-2)$$

$$=H(t-2)(-H(t-2)-(t-2)+e^{t-2})$$

$$L^{-1}(\dfrac{e^{-2s}}{s^2(s-1)})=H(t-2)(-H(t-2)-(t-2)+e^{t-2})$$