Concept explainers
Find the time-varying function
Answer to Problem 55P
The time-varying function
Explanation of Solution
Given data:
The differential equation is,
The initial conditions are zero. That is,
Formula used:
Write the general expression for the Laplace transform.
Write the general expression for the inverse Laplace transform.
Write the general expressions to find the Laplace transform function.
Here,
Write the general expressions to find the inverse Laplace transform function.
Calculation:
Apply Laplace transform function given in equation (2), (4), (5), (6) and (7) to equation (1).
Substitute 0 for
Rearrange the equation (13) to find
Reduce the equation as follows,
Expand
Here,
A, B, C, D and E are the constants.
Now, to find the constants by using residue and algebraic method.
Constant A:
Substitute equation (14) in equation (16) to find the constant A.
Constant B:
Substitute equation (14) in equation (17) to find the constant B.
Constant C:
Substitute equation (14) in equation (18) to find the constant C.
Consider the partial fraction,
Reduce the equation as follows,
Equate the coefficients of
Substitute
Equate the coefficients of
Substitute
Substitute
Reduce the equation as follows,
Apply inverse Laplace transform given in equation (3) to equation (22). Therefore,
Apply inverse Laplace transform function given in equation (8), (9), (10) and (11) to equation (23).
Conclusion:
Thus, the time-varying function
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