Hello and thank you for posting your question to Brainmass!
The solution is attached below in two files. ...

Solution Summary

Fourier transform is a powerful tool when used to solve ordinary and partial differential equations. This solution shows step by step how to go from the initial equation, utilizing the Fourier transform, to the inverse transform and complex contour integration to reach a solution (in an integral form). The solution contains 4 pages with full derivations and is provided in both a pdf. and Word document.

Please see attachment.
1. What is the FourierTransform for the convolution of sin(2t)*cos(2t).
2. Compute the inverse Fouriertransform for X(w)= sin^2*3w
3. A continuous time signal x(t) has the Fouriertransform
X(w) = 1/jw+b where b is a constant. Determine the Fouriertransform for v(t) = x*(5t-4)

Please see the attached file for details.
1. A continuous time signal x(t) has the Fouriertransform X(w) = 1/(jw + b), where b is a constant. Determine the Fouriertransform for v(t) = x(5t - 4).
2. For a discrete-time signal x[n] with the DTFT X(w) = 1/(e^jw + b), where b is an arbitrary constant compute the DTFT V(Ω

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

The problem is from Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform:
Solve using a cosine or sine transform.
u'' - 9u =50e^-3x (0

Show that integral from - infinity to + infinity of psi_1(x) times psi_2*(x) dx is equal to integral from - infinity to + infinity of phi_1(k) times phi_2*(k) dk.
* indicates complex conjugate

A waveform v(t) has a Fouriertransform which extends over the range -F to +F in the frequency domain. The square of the waveform v(t), that is, v(t) v(t), then has a Fouriertransforms which extends over the range:
==============================================
Answers:
a) -F to +F
b) -2F to +2F
c) -3F to +3F
d) -4F to +

Please show all steps.
1. Let f(x) be a 2pi- periodic function such that f(x) = x^2 −x for x ∈ [−pi,pi].
Find the Fourier series for f(x).
2. Let f(x) be a 2pi- periodic function such that f(x) = x^2 for x ∈ [−1,1]. Using
the complex form, find the Fourier series of the function f(x).
3. See attachmen