Question

Determine the inverse Laplace Transform of

(2s+8)∗exp(−5s)/s²+8s+15. Show your steps. Fill in the coefficients of the inverse Laplace Transform in the form of a*exp(_______)*exp(_______t)+b*exp(________)*(exp(_______t), where a and b are some specific constants that you do not need to report. You only need to fill in the four blanks. Write the first term for the pole with a smaller magnitude (absolute value), and the second term for the pole with a bigger magnitude. If the term is positive, there is no need to write + . If the term is negative, please remember to add - in the front.

Answer 1

To find the inverse Laplace Transform of (2s+8)*exp(−5s)/s²+8s+15, we perform partial fraction decomposition and apply the inverse Laplace transform, considering poles at s=-3 and s=-5. The exponential delay factor from exp(−5s) is accounted for in the final expression. The blanks are filled with exponents resulting in a∗exp(5)∗exp(-3t) + b∗exp(5)∗exp(-5t).

Step-by-step explanation:

To determine the inverse Laplace Transform of (2s+8)∗exp(−5s)/(s²+8s+15), we first decompose the given expression into partial fractions. The quadratic in the denominator factors as (s+3)(s+5), revealing the poles of the transform at s=-3 and s=-5. However, given the exp(−5s) term, we need to consider the shift theorem, which implies that each term will be multiplied by an exponential of the form exp(5t) in the time domain. This factor accounts for the delay introduced by exp(−5s).

The partial fraction decomposition yields coefficients A and B, such that:

∗ A/(s+3) + B/(s+5)

Applying the inverse Laplace transform to each partial fraction term:

∗ A exp(-3t) + B exp(-5t)

With the exponential delay, the final inverse Laplace equation is thus:

∗ A∗exp(5t)∗exp(-3t) + B∗exp(5t)∗exp(-5t)

Since we are asked to report the exponents only:

∗ a∗exp(_______)∗exp(_______t) + b∗exp(________)∗exp(_______t)

Matching the terms to the expression given in the question:

∗ a∗exp(5)∗exp(-3t) + b∗exp(5)∗exp(-5t)

Therefore, the blanks in the given expression should be filled in as follows:

a∗exp(5)∗exp(-3t) + b∗exp(5)∗exp(-5t)