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Determine the inverse Laplace transform of the function below.

5s+39
/s
2
+14s+53



1 Answer

3 votes

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)

User Henrique Bastos
by
7.7k points

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