
Concept explainers
(a)
Find the inverse Laplace transform for the function
(a)

Answer to Problem 30P
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
The Laplace transform function is,
Formula used:
Write the general expression for the inverse Laplace transform.
Write the general expression to find the inverse Laplace transform function.
Here,
Calculation:
Expand
Here,
A, B, and C are the constants.
Now, to find the constants by using algebraic method.
Consider the partial fraction,
Reduce the equation as follows,
Equating the coefficients of
Equating the coefficients of
Equating the coefficients of constant term in equation (7).
Substitute equation (10) in equation (8) to find the constant B.
Substitute equation (10) in equation (9) to find the constant C.
Substitute
Reduce the equation as follows,
Apply inverse Laplace transform given in equation (2) to equation (11). Therefore,
Apply inverse Laplace transform function given in equation (3), (4) and (5) to equation (12).
Conclusion:
Thus, the inverse Laplace transform for the given function is
(b)
Find the inverse Laplace transform for the given function
(b)

Answer to Problem 30P
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
The Laplace transform function is,
Formula used:
Write the general expressions to find the inverse Laplace transform function.
Calculation:
Expand
Here,
D, E, and F are the constants.
Now, to find the constants by using residue method.
Constant D:
Substitute equation (13) in equation (17) to find the constant D.
Constant E:
Substitute equation (13) in equation (18) to find the constant E.
Constant F:
Substitute equation (13) in equation (19) to find the constant F.
Substitute
Apply inverse Laplace transform given in equation (2) to equation (20). Therefore,
Apply inverse Laplace transform function given in equation (14) and (15) to equation (21).
Conclusion:
Thus, the inverse Laplace transform for the given function is
(c)
Find the inverse Laplace transform for the function
(c)

Answer to Problem 30P
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
The Laplace transform function is,
Calculation:
Expand
Here,
A, B, and C are the constants.
Now, to find the constants by using algebraic method.
Consider the partial fraction,
Reduce the equation as follows,
Equating the coefficients of
Equating the coefficients of
Equating the coefficients of constant term in equation (24) on both sides.
Substitute equation (25) and (27) in equation (26) to find the constant A.
Rearrange the equation as follows,
Substitute
Substitute
Substitute
Reduce the equation as follows,
Apply inverse Laplace transform given in equation (2) to equation (28). Therefore,
Apply inverse Laplace transform function given in equation (3), (4) and (15) to equation (29).
Conclusion:
Thus, the inverse Laplace transform for the given function is
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Chapter 15 Solutions
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