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example.problems.midterm2.SOLUTIONS - 1 Find the inverse...

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Unformatted text preview: 1. Find the inverse Laplace transform of Solution Note that . . Using Table 4.1 in the textbook, . 2. Find the inverse Laplace transform of Solution Note that , thus 3. Compute the transfer function corresponding to the differential equation Solution The use of a transfer function implicitly assumes that initial conditions are zero. Thus, . The transfer function is 4. Is the system in Problem 3 stable? Solution The denominator is factored , thus the poles of the transfer function are and . 5. Compute the output of the system in problem 3 when the input is a unit step . Solution The Laplace transform of the output is which through partial fraction expansion is equal to Thus 6. Compute the steady‐state output of the transfer function in Problem 3 when the input is a unit step . Hint: use the final value theorem. Does your answer match what you get by simply computing the limit of the answer in Problem 5 as ? Solution Since the transfer function is stable we can use the final value theorem, 7. Compute the inverse Laplace transform of Solution When the transfer function is proper, but not strictly proper, we expand into a whole term and a fractional term. That is, we write . Thus where we compute and is to derive the fact that the 8. Use the fact that the Laplace transform of Laplace transform of is Solution Let , then . The Laplace transform of is , and the Laplace transform of is . Thus This is the same answer that we get by computing ...
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