Question

The inverse Laplace transformation of \( \frac{s+5}{s^2 + 4s + 4} \) for \( t \geq 0 \), is

• A. \( 4e^{2t} \)
• B. \( 4e^{-2t} \)
• C. \( (1 + 3t)e^{2t} \)
• D. \( (1 + 3t)e^{-2t} \)

Hint:

When simplifying rational functions in Laplace transforms, always attempt to factor the denominator and apply known inverse Laplace transforms directly.

Answer 1

The Correct Option is D

We are given the function \( \frac{s+5}{s^2 + 4s + 4} \). To begin, we observe that the denominator can be factored as \( (s+2)^2 \). 
This simplifies the expression to \( \frac{s+5}{(s+2)^2} \). 
Now, we perform partial fraction decomposition or use a standard inverse Laplace table. 
Recognizing this as a standard form for inverse Laplace, we know the inverse transform of \( \frac{s+2}{(s+2)^2} \) is \( e^{-2t} \), and the additional constant factor of 3t leads us to the final answer of \( (1 + 3t)e^{-2t} \).