Final answer:
To obtain the inverse Laplace transform x(t) for each of the given transforms, we need to decompose the given rational functions into partial fractions and then find the inverse Laplace transform of each term.
Step-by-step explanation:
To obtain the inverse Laplace transform x(t) for each of the given transforms, we need to decompose the given rational functions into partial fractions and then find the inverse Laplace transform of each term.
a. X(s) = 3 / s(s+2)
To decompose the rational function, we express X(s) as a sum of two terms: X(s) = A/s + B/(s+2).
By comparing the coefficients, we find A = 2 and B = -2.
Therefore, the inverse Laplace transform of X(s) is x(t) = 2 - 2e^(-2t).
b. X(s) = (10s + 7) / s(s+3)
Using partial fractions, we express X(s) as X(s) = A/s + B/(s+3).
By comparing the coefficients, we find A = 7/3 and B = 7/3.
Therefore, the inverse Laplace transform of X(s) is x(t) = (7/3)e^(-3t) + (7/3)e^(-2t).
c. X(s) = (4s + 7) / (s+2)(s+5)
Using partial fractions, we express X(s) as X(s) = A/(s+2) + B/(s+5).
By comparing the coefficients, we find A = -1/3 and B = 7/3.
Therefore, the inverse Laplace transform of X(s) is x(t) = (-1/3)e^(-2t) + (7/3)e^(-5t).
d. X(s) = 5 / s²(s+4)
Using partial fractions, we express X(s) as X(s) = A/s + B/s² + C/(s+4).
By comparing the coefficients, we find A = 1, B = -4, and C = 3.
Therefore, the inverse Laplace transform of X(s) is x(t) = 1 - 4t + 3e^(-4t).
e. X(s) = (3s+2) / s²(s+4)
Using partial fractions, we express X(s) as X(s) = A/s + B/s² + C/(s+4).
By comparing the coefficients, we find A = 1/4, B = -3/4, and C = 1/2.
Therefore, the inverse Laplace transform of X(s) is x(t) = (1/4) - (3/4)t + (1/2)e^(-4t).
f. X(s) = (12s+5) / (s+3)²(s+7)
Using partial fractions, we express X(s) as X(s) = A/(s+3) + B/(s+3)² + C/(s+7).
By comparing the coefficients, we find A = -1/40, B = 1/20, and C = 1/40.
Therefore, the inverse Laplace transform of X(s) is x(t) = (-1/40)e^(-3t) + (1/20)te^(-3t) + (1/40)e^(-7t).