Accurately sketch the spectrum (magnitude only) of the following function y(t)=x(t) \cos (2 \pi 10 t) \text { if } x(t)=3 \operatorname{rect}\left(\frac{t}{2}\right)

4. Solve the initial value problem ਬਚਤਰ ਹੈ। 이 rigk |||| 3r-z -3y-z 2y-2 Bou dt with z(0) = -5, y(0) = 13, 2(0) = -26 using eigenvalue/eigenvector techniques.

3. Consider the system tx' = [22]x, t> 0 with initial condition X(2) = [−12]. A Assuming solutions of the form x = where X, V are an eigenvalue/eigenvector pair of the given matrix, use techniques similar to those used to construct solutions to the constant coefficient linear homogeneous systems to solve the given initial value problem. Write your answer as a single vector.

Calculate the response of the system modeled by the equation (units are in Newtons): 5x + 125x = 108(t) xo= 1 m, vo= -5 m/s

5 Use Parseval's identity to compute the following integrals. \text { a) } \int_{-\infty}^{\infty} \operatorname{sinc}^{2}(2 r) d t \text {, } \text { (b) } \int_{0}^{\infty} \operatorname{sinc}(t) \operatorname{sinc}(2 t) d t \text {. }

Find the eigenvalues of Hint: You may prefer to use row operations to find the characteristic polynomial.

a. find the eigenvalues (real or complex) of, b.what are the eigenvalues of (M-121)?

6. Tell whether the systems 5(a) and 5(b) memoryless /Not-memoryless, causal/non-causal?Show proof