Answer: To find the answer we can use a partial function decomposition.
Step-by-step explanation: Find the denominator of the expression
S^2 + 14s + 53 = ( s + 7 )(s +7)
Which gives us
(5s + 39)/((s+7)(s+7)
Express the expression as the sum of two fractions:
(5s + 39)/((s+7))=A/(s+7)+B/(s+7)^2
To find A and B equate the numerator on the left side to the sum of the numerators on the right side
5s + 39= A(s+7)+B
expanding and simplifying
5s+39=As+7A+B
Equate the coefficients of like terms
5=A
39=7A+B
From the first equation we find that A=5 substituting this into the second equation
39=7(5)+B
39=35+B
B= 39-35
B=4
rewrite
(5s+39)/((s+7)(s+7))=5/(s+7)+4/(s+7)^2
The inverse Laplace transform of 5/(s+7) is 5e^(-7t), and the inverse Laplace transform of 4/(s+7)^2 is 4te^(-7t)
therefore the inverse Laplace transform of (5s+39)/(s^2+14s+53) is 4e^(-7t) + 4te^(-7t)