Answer to Question #126618 in Differential Equations for Ibrahim

Question

Solve
L^-1 × ((s^2 - 5s - 7)/(s + 2)^3)

Answer 1

Given,

$$\mathscr{L^{-1}} \bigg(\dfrac{s^2 - 5s - 7}{(s + 2)^3}\bigg)$$

Thus,

$$\implies\mathscr{L^{-1}}\left\{\frac{1}{s+2}-\frac{9}{\left(s+2\right)^2}+\frac{7}{\left(s+2\right)^3}\right\}$$ $$\implies \mathscr{L^{-1}}\left\{\frac{1}{s+2}\right\}-\mathscr{L^{-1}}\left\{\frac{9}{\left(s+2\right)^2}\right\}+\mathscr{L^{-1}}\left\{\frac{7}{\left(s+2\right)^3}\right\}$$

$$\implies e^{-2t}-e^{-2t}\cdot \:9t+\frac{7e^{-2t}t^2}{2}$$