Relating Transform of a Function and Transform of the Derivative

Please see the attached file for the fully formatted problems.

Problem statement:
What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will prove below is key to all that follows.

To do the following problem, you will need the definite integral form of the integration by parts formula:

Let be a function such that (you'll need this assumption in the following). Use integration by parts to show the following about Laplace transforms of derivatives.
Hint: makes an excellent .

OK, so I've said so far that
I let my u = and let dv = , du = -s and v = but then I get confused about how to implement the definite integral form of integration by parts, and won't I have to use this integration by parts several times?

After getting the integration by parts finished, I know I will have to take the limit as b approaches infinity from 0 to b. Then I can maybe apply L'Hopital's rule to simplify and prove that ?

Please see the attached file for the complete solution.
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Problem statement:
What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will ...

Solution Summary

The Laplace transform of a function and its derivative are investigated. The solution is detailed and well presented.

I have attached some problems that I think I am working correctly. I wanted to verify the concept is correct.
A signal x(t) = exp(-t)*cos(3t) is turned on at t = 0. What is its Fourier transform?
Consider the signal in problem 6. What is the Fourier transform of its derivative with respect to time?
What is the Fourie

1. Calculate the Laplace transform of exp(-10t) x u(t)
2. Calculate the convolution of exp(-t) and sin(t) using (a) the Laplace transform (b) direct integration
3. Compute the inverse transform of (3s^2 + 4s + 1) / (s^4 + 3s^3 +3s^2+2s)
4. Use Laplace transform to calculate the solution to the ODE y"+6y'+8y=u(t) y

Attached are three problems that I am working. Any assistance would be greatly appreciated.
For a discrete-time signal x[n] with the z-Transform:
X(z) = z
________________________________________8z2-2z-1
find the z-Transform, V(z) for the signal v[n] = e3nx[n].
See attached for the rest of the

Please see the attached file and include an explanation of problem. Thank you.
1. Compute the Fourier transform for x(t) = texp(-t)u(t)
2. The linearity property of the Fourier transform is defined as:
3. Determine the exponential Fourier series for:
4. Using complex notation, combine the expressions to form a singl

For the following expression:
x(n) = 3 (0.75)^n cos(0.3 pi n)u(n) + 4 (0.75)^n sin(0.3 pi n) u(n),
Please calculate and express the Z transform as a rational function Z^-1 and work out the ROC.

Laplace Transform
Application of Complex Inversion Integral Formula
(Bromwich's Integral Formula)
Problem:- Find the Laplace Transform of thefunction F(t) = (1 - e^(

Consider computing the Laplace transform of a pulse:
p(t) = {1 0 < t < 1
{0 otherwise
a) use the integral formula to find P(s), the Laplace transform of p(t). Determine the region of convergence of P(s).
b)Represent p(t) in terms of the unit-step functionand use its Laplace transformandthe time-shift p