Answer to Question #208682 in Calculus for Didues

Question

L^-1(s² - 4 / ( s+2) ²)

Answer 1

Using the table of originals and images, we get:

${L^{ - 1}}\left( {\frac{1}{{{{(s + 2)}^2}}}} \right) = t{e^{ - 2t}}$

By the image differentiation theorem

${L^{ - 1}}\left( {{s^2}} \right) = \delta ''\left( t \right)$

Then

${L^{ - 1}}\left( {{s^2} - \frac{4}{{{{(s + 2)}^2}}}} \right) = {L^{ - 1}}\left( {{s^2}} \right) - 4{L^{ - 1}}\left( {\frac{1}{{{{(s + 2)}^2}}}} \right) = \delta ''\left( t \right) - 4t{e^{ - 2t}}$

Answer: $\delta ''\left( t \right) - 4t{e^{ - 2t}}$ , $\delta \left( t \right)$ is Dirac delta function