Inverse Laplace transform of $(2s+4)/(s^2+4s+5)^2$

Question

Out of many transformation shortcuts in Laplace table I still find difficulty in finding the inverse laplace transform of $\displaystyle \frac{2s+4}{(s^2+4s+5)^2}$. I tried partial fraction and its very difficult. Completing the square did not work for me. I hope someone can help me. Thanks.

Answer 1

HINT: Try completing the square in the denominator and playing with the numerator (expressing the fraction into two or more fractions if needed) to see if you can find any familiar inverse Laplace Transform functions.

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

Can you search the Laplace Table to find something that looks really familiar?

Like...

$$L[e^{ct}t\sin at]=\frac{2a(s-c)}{((s-c)^2+a^2)^2}$$

Seems like I've given you the answer already. Heh.