1. For the waveform shown in FIGURE 1 (see attached file), estimate:
(a) the damping factor (you may compare response with a standard chart)
(b) the forced or damped frequency of oscillation
(c) the natural or undamped frequency

2. Complete the following inverse Laplace transforms (see attached files)

This solution provides an analysis of a given underdamped waveform to determine the natural frequency, damped frequency and damping factor using peak values from the waveform and logarithmic decrement analysis tools. The second part shows solutions of some inverse laplace transform examples including some using partial fraction expansion to deriuve the standard inverse Laplace transform representation

Please show all the steps and working.
1. Complete the following inverseLaplacetransforms:
a) 1/(2s + 4)
b) 5s/(16+s^2)
c) 10/s^2
d) 4/(s-2)^2 + 16
e) (9s+ 23) /(s+2)(s+3)
2. Complete the following Laplacetransforms:
a) 20
b) 20sin(5t)
c) 20exp(-5t)
e) 20exp(-5t)cos(5t)
f) 20t^2
Please see the attachment fo

Please walk me through these problems step by step.
1. Compute the Laplace transform of e^(-10t) = u(t).
2. Compute the inverseLaplace transform of:
X(s) = (3s^2+2s+1) / (s^3+5s^2+8s+4)
3. Use Laplacetransforms to compute the solution to the differential equation given in the attachment.

I have a transform F(s) of which I need the inverse transform for. The form of the transform is not of a common form and I am having trouble reducing it to a workable form.
I am looking at a problem that requires the inverselaplace transform of f(t) to be found using the following transform:
F(s) = (s*e^(-s/2))/(s^2 + p

1. Find the inverse of F(s) = s/[(s+1)(s^2+4)] the answer can be left as a convolution integral
2.Solve y''+y'+(5/4)y=y(t) y(0)=y'(0)=0 using LaPlaceTransforms
sin(t) t greater or equal to 0 or less than Pi
y(t) {
0 t greater or equal than Pi

Find the InverseLaPlace Transform using different methods described in the attachment.
To see the description of the problem in its true format, please download the attached question file.

Consider the system (see attached file). Determine the transfer function of the system in terms of Laplace transform. Expand the transfer function by means of partial fractions and find your new state equations by means of inverseLaplace transform. Obtain the state-space representation of the system.

(1) Use LaplaceTransforms to solve Differential Equation
y'' - 8y' + 20 y = t (e^t) , given that y(0) = 0 , y'(0) = 0
(2) Use LaplaceTransforms to solve Differential Equation
y''' + 2y'' - y' - 2y = Sin 3t , given that y(0)=0 , y'(0)=0 ,y''(0)=0, y'''(0)=1
Note: To see the questions in their mathematic