For the following problems you may use the properties of the Laplace Transform (link to table), and the Laplace transform of an exponential.

No integration should be necessary.

a) Derive the Laplace Transform of a sinusoid (you needn't do any integrals to solve this one). Recall Euler's identity for sine.

b) Derive the Laplace Transform of a decaying sinusoid (you needn't do any integrals to solve this one).

c) Given the Laplace Transform of the sine function

use the differentiation property of the Laplace transform to find the Laplace transform of the cosine. In other words show that wwithout taking the derivative in the time domain.

a)

b)

... or use complex shift property

c)

Derive the convolution property of the Laplace Transform (assume f_{1}(t)
and f_{2}(t) are causal (i.e., they are zero for t<0)). The
convolution property states that the Laplace Transform of the convolution of
two function is simply the product of the Laplace Transforms of the
functions - this makes convolution easy.

Hint: Write the definition of the convolution (in terms of integrals) and then use it in the definition of the Laplace Transform. This will give you a double integral - you can switch order of integration and with some judicious change of variables (and pulling constants out of integrals as appropriate) the answer develops.

See here for explanation of the steps in the solution

For each of the examples below y(t)=f(t)*h(t) (y is the convolution of f and h). All functions are implicitly zero for t<0.

- Write the convolution integral in time (don't solve it - it is just there to remind you how much easier these problems are in t).
- Find F(s), H(s) and Y(s)
- Find y(t)

a)

b)

c)

d)

e)

Note: in all of your solutions there was no guessing forms of solutions, or hard integration. In particular, part e would require integration by parts.

a)

b)

c)

d)

e)

The average of a function over the previous T_{0} seconds is given
by:

Find an expression for Y_{avg}(s) in terms of F(s) and T_{0}.

a) Draw the free body diagram for the system shown. The mass rolls on frictionless, massless wheels.

b) Write the differential equation with x_{in}(t) as input and_{}x_{out}(t) as output.

c) Use Laplace Transforms to find the impulse response of the system if m=1, b=3, k=2. Remember the impulse response is a zero state response.

d) What is the impulse response of the system if m=1, b=2, k=4?
Hint: try completing the square so that

s^{2}+2s+4=(s+1)^{2}+3

e) What is the impulse response of the system if m=1, b=2, k=1?

f) Identify the systems in c, d, and e as under-, over-, and critically damped.

Note that you were able to get all of the impulse responses with no guessing forms of solution, or integrating from 0- to 0+ or finding the step response and differentiating. Laplace makes much work with differential equations easier.

a)

b)

c)

Inverse Transform was used done with "double exponential" entry in
table.

Characteristic equation is s^{2}+3s+2 has two real roots.

d)

We need a term equal to 3^{½} in the numerator to get into
form that is in the table

Inverse Transform was used done with "decaying sine" entry in
table.

Characteristic equation is s^{2}+2s+4 has complex conjugate roots.

e)

Inverse Transform was used done with "time multiplied exponential" entry in
table.

Characteristic equation is s^{2}+2s+1 has a repeated real root.

f)

Part (c) has two real distinct exponentials: overdamped.

Part
(d) has complex exponential (or decaying sinusoid): underdamped.

Part
(e) has an exponential and a time-multiplied-exponential: critically damped.