Understand forward and inverse laplace transform and obtain

Question

Homework

The next problem(s) addresses the following course learning objective(s):

- Understand forward and inverse Laplace transform.
- Obtain the initial value and final values using Laplace transform.
- Obtain a transfer function using the Laplace transform.

Problem 1

Use the defining integral to

a) find the Laplace transform of cosh βt;

b) find the Laplace transform of sinh βt.

Problem 2

Use the appropriate operational transform from Table 12.2 to find the Laplace transform of each function:

a) t²e^-at;

b) d/dt(e^-at sinh βt);

c) t cos ωt.

Problem 3

Find f (t) if

F(s) = 6s² + 26s + 26/((s + 1)(s + 2)(s + 3))

Problem 4

Find f(t) if

F(s) = (10(s² + 119))/((s + 5)(s2 + lOs + 169))

Problem 5

Find f (t) if

F(s) = 40/(s² + 4s + 5)².

Problem 6

Find f (t) if

F(s) = (2s³ + 8s² + 2s - 4)/(s2 + 5s + 4)

Problem 7

Using the initial- and final-value theorems, find the initial and final values of f(t) given as below:

a). F(s) = (7s² + 63s + 134)/(s + 3)(s + 4)(s + 5)

b). F(s) = (4s² +78 +1)/ s(s + 1)²

c). F(s) = 40/(s² + 4s + 5)²

Problem 8

The switch in the circuit in Fig. P12.30 has been in position a for a long time. At t = 0, the switch moves instantaneously to position b.

a) Derive the integrodifferential equation that governs the behavior of the current i o for t ≥ 0⁺.

b) Show that

I_o(s) = - I_dc [s + (1/RC)]/[s² + (1/RC)s + (1/LC)].