Concept explainers
(a)
Find the inverse Laplace transform for the given function
(a)
Answer to Problem 31E
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
Consider the Laplace transform function is,
Formula used:
Write the general expression for the inverse Laplace transform.
Calculation:
Expand
Here,
A, B, and C are the constants.
Find the constants by using algebraic method.
Consider the partial fraction,
Put
Put
Put
Subtract equation (5) and (6),
Substitute 1 for C in the above equation.
Substitute
Substitute
Apply inverse Laplace transform of equation (2) in equation (8).
Write the general expression to find the inverse Laplace transform function.
Write the general expression to find the inverse Laplace transform function.
Apply inverse Laplace transform function of equation (9) and (10), in equation (8).
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 31E
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
Consider the Laplace transform function is,
Calculation:
The equation (11) can be rewritten as follows,
Expand
Here,
A, B, C are the constants.
Find the constants by using algebraic method.
Consider the partial fraction,
Put
Expanding equation (14) as follows,
Substitute
Equating the coefficient of
Equating the coefficient of constant term in equation (15).
Susbtitute 0 for A in the above equation.
Substitute 0 for A, 1 for B, and
Write the general expression to find the inverse Laplace transform function.
Apply inverse Laplace transform function of equation (10) and (17), in equation (16).
Conclusion:
Thus, the inverse Laplace transform for the given function is
(c)
Find the inverse Laplace transform for the given function
(c)
Answer to Problem 31E
The inverse Laplace transform for the given function is
Explanation of Solution
Given data:
Consider the Laplace transform function is,
Calculation:
Expand
Here,
A, B, C, and D are the constants.ind the constants by using algebraic method.
Consider the partial fraction,
Substitute
Substitute
Substitute
Apply inverse Laplace transform of equation (2) in equation (21).
Apply inverse Laplace transform function of equation (9) in equation (22).
Conclusion:
Thus, the inverse Laplace transform for the given function is
(d)
Verify the functions given in Part (a), Part (b), and Part (c) with MATLAB.
(d)
Answer to Problem 31E
The given functions are verified with MATLAB.
Explanation of Solution
Calculation:
Consider the function given in Part (a).
The MATLAB code for the given function:
syms s t
ilaplace(1/(s+2)/(s+2)/(s+1))
MATLAB output:
Consider the function given in Part (b).
The MATLAB code for the given function:
syms s t
ilaplace(s/(s+2)/(s+2)/(s+2))
MATLAB output:
Consider the function given in Part (c).
The MATLAB code for the given function:
syms s t
ilaplace(1/(s*s+8*s+7))
MATLAB output:
Conclusion:
Thus, the given functions are verified with MATLAB.
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Chapter 14 Solutions
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