Answer to Question #343134 in Differential Equations for Tobias Felix

Question

Inverse Laplace Transforms

Find L^-1 {F(s)} when F(s) is given by

4. s+7/s^2+2s+5

Answer 1

4.

$$\dfrac{s+7}{s^2+2s+5}=\dfrac{s+1+6}{(s+1)^2+4}$$

$$=\dfrac{s+1}{(s+1)^2+2^2}+6\dfrac{1}{(s+1)^2+2^2}$$

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

$$+6L^{-1}(\dfrac{1}{(s+1)^2+2^2})=e^{-t}\cos(2t)+\dfrac{6}{2}e^{-t}\sin (2t)$$

$$=e^{-t}\cos(2t)+3e^{-t}\sin (2t)$$