Find the general solution to the homogeneous equation find

Question

Assignemnt

You may use your books and notes during the test. You may use your laptop if your book is electronic, but you are on your honor not to use your laptop for any other purpose.

1. Use Laplace transform techniques to solve the following initial value problem. (This problem is trivial, but I want you to demonstrate that you know how to use the transform technique. Be sure to include all the important steps.)

y^· + y = 0, y(0) = 1

2. Perform the convolution h_k * u_k, assuming both are causal.

h_k = (1/4)^k - 1, u_k = (1/2)^k

3. Consider the following discrete time system. (Hint: All the roots of the auxiliary equation are real and distinct. A good trick is to try factors of the lowest order term to see if they are roots. Please don't waste your time using the cubic formula.)

y_k - 6y_k-1 11y_k-2 - 6y_k-3 = U_k

a) Find the general solution to the homogeneous equation.
b) Find the impulse response sequence.
c) Find the particular solution for u_k = 2^k.

4. Find the inverse impulse response sequence h^{^}_k for the system with impulse response sequence

h_k = (1/3)^k - 1/2