The inverse Laplace transform of the function ${{s + 5} \over {\left( {s + 1} \right)\left( {s + 3} \right)}}$ is $2e^{-t} – e^{-3t}$.
To find the inverse Laplace transform, we can use the partial fraction expansion:
$${{s + 5} \over {\left( {s + 1} \right)\left( {s + 3} \right)}} = {{A} \over {s + 1}} + {{B} \over {s + 3}}$$
To find $A$ and $B$, we multiply both sides of the equation by $\left( {s + 1} \right)\left( {s + 3} \right)$:
$$s + 5 = A(s + 3) + B(s + 1)$$
Substituting $s = -1$, we get $A = 2$. Substituting $s = -3$, we get $B = -1$.
Therefore, the inverse Laplace transform is:
$${{s + 5} \over {\left( {s + 1} \right)\left( {s + 3} \right)}} = {{2} \over {s + 1}} + {{-1} \over {s + 3}}$$
The inverse Laplace transform of ${{2} \over {s + 1}}$ is $2e^{-t}$, and the inverse Laplace transform of ${{-1} \over {s + 3}}$ is $-e^{-3t}$. Therefore, the inverse Laplace transform of ${{s + 5} \over {\left( {s + 1} \right)\left( {s + 3} \right)}}$ is $2e^{-t} – e^{-3t}$.
Here is a brief explanation of each option:
- Option A: $2e^{-t} – e^{-3t}$ is the correct answer.
- Option B: $2e^{-t} + e^{-3t}$ is incorrect because it does not include the factor of $-1$ in the denominator.
- Option C: $e^{-t} – 2e^{-3t}$ is incorrect because it does not include the factor of $2$ in the numerator.
- Option D: $e^{-t} + 2e^{-3t}$ is incorrect because it does not include the factor of $-1$ in the denominator.