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.

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.

5. Consider the system x' following values of c: (a) c = (0,00) 9 x where c is a parameter. Classify the geometry and stability properties of the system for the (b) c = 0 (c) c = (-1,0) (d) c= -1 (e) c = (-∞, -1)

esc ZILLDIFFEQ9 10.2.010. Classify the critical point (0, 0) of the given linear system by computing the trace r and determinant A and using the figure. X'= -9x + 3y y' = 6x - 5y O center Stable node unstable node O stable spiral O unstable spiral Osaddle Ostable node Degenerate stable node O degenerate unstable node O degenerate stable node PREVIOUS ANSWERS 2 Stable spiral 80 44 Unstable spiral T²-44 <0 Center Saddle T²=4A Unstable node Degenerate unstable node ZE DII il later. $19 be downloaded. Please try again MY NOTES F12

2. Use the following figure to construct a model for the number of pounds of salt ₁ (t), ₂(t), and z3(t) at time t in tanks A, B, and C, respectively. Write the model in matrix form and then solve it using eigenvalue/eigenvector techniques assuming that 21 (0) = 15, 22(0) = 10, and 23 (0) = 5. Will all of the tanks eventually be free of salt? Use your solution to justify your answer. pure water 4 gal/min 200 gal mixture 4 gal/min 150 gal mixture 4 gal/min امع 1000 mixture 4 gal/min

A vibrating system is described by the equation (units are in Newtons): 5x + 125x = 10 cos 5t Compute the response of the system if the system is initially at rest.

1.Consider the coupled acoustic-mechanical system. The velocity of the masses are given by u and applied force by the variable f. The variables k represent the mechanical stiffness, M the mass and b the damping coefficient. The closed open pipe is filled with a fluid having mass density Po, sound speed c,cross sectional area A , length L. а. Using mobility analogy where the velocity as the "across" variable, determine the an equivalent circuit for the system. b. Determine the equations of motion in the Laplace-domain. c. Determine the equations of motion in the time-domain. d. Find the transfer function U2(s)/Uo(s).

Consider the following linear program: \begin{array}{l} \max x_{1}+x_{2} \\ \text { s.t.: } \\ \text { Cl } 3 x_{1}+2 x_{2} \leq 12 \\ \text { C2 } 2 x_{1}+3 x_{2} \leq 12 \\ \text { C3 } 2 x_{1}+2 x_{2} \leq 9 \\ \text { C4 } 2 x_{1}+2 x_{2} \geq 3 \\ \text { CS } \quad x_{1}, x_{2} \geq 0 \end{array} a) (5 points) Graph the feasible region of the LP. Is the feasible region unbounded? b) (35 points) Solve this problem using Simplex algorithm. Make sure to indicate: * Whether or not you need to perform Phase * Show the BV and NBV for each iteration of Simplex * Show the steps of the algorithm in the graph from point a) * Is this a unique or multiple solution?

. Obtain the Fourier Transform of the following functions \text { a. } x(t)=\left\{\begin{array}{l} e^{a t}, t<0 \\ e^{-a t}, t>0 \end{array}\right\} b. Signal shown below:

A signal x(t) has a Fourier Transform given by X(\omega)=\frac{5(1+j \omega)}{8-\omega^{2}+6 j \omega} Without finding x(t), find the Fourier Transform of the following: а. х(t-3) b. x(4t) C. ei1.e12x(t) d. x(-2t)

1. (30 pts) For each of the following systems, determine whether it is linear and whether it is time-invariant. Justify your answers. If it is LTI, find the impulse response function h(t). Each system is specified by the output y that is produced from an input r. \text { (a) } y(t)=x(t+7) \text { (b) } y(t)=x(3 t) \text { (c) } y(t)=|x(10)| y(t)=\int_{-\infty}^{\infty} I_{[0,+\infty)}(t-\tau) \exp (\tau-t) x(\tau) d \tau y(t)=\int_{-\infty}^{\infty} \frac{1}{1+\tau^{2}} x(\tau-t) d \tau y(t)=\int_{-1}^{0}(\tau-1) x(t+\tau) d \tau y(t)=\min (1, \max (-1, x(t-4))) n) Let (a1,. , ak) be a vector of k nonnegative reals and let (T1,.., Tk) E R*. y(t)=\underset{x \in \mathbb{R}}{\operatorname{argmin}} \sum_{i=1}^{k} a_{j}\left(z-x\left(t-\tau_{i}\right)\right)^{2} The argmin, is the value of z (the argument) that minimizes the expression.

2. Let s be a periodic signal with period To = 2 and s(t)=\left\{\begin{array}{ll} -t(t-1) & 0 \leq t<1 \\ (t-1)(t-2) & 1 \leq t<2 \end{array}\right. ) Find the first, second, and third derivatives r = Ds, u= D²s, and v = D³s. s) Find the Fourier coefficients of each of the four signals: §, î, û, and û. :) (12 pts) For each of the four signals, compute the power with a time domain calculation and compute the power in frequencies ±1/2 (the positive and negative fundamental frequencies) with a frequency domain calculation. What fraction of the power is in the positive and negative fundamental frequencies? Express all answers both symbolically and with an approximate decimal representation.

4 Find and sketch the Fourier transforms for the following signals. u(t)=(1-|t|) I_{[-1,1]}(t) v(t)=\operatorname{sinc}(2 t) \operatorname{sinc}(4 t) \text {. } \text { (c) } s(t)=v(t) \cos (200 \pi t) \text {. } (d) Classify cach of the signals in (a) (c) as baseband or passband.

4. Determine which of the following systems is linear or non-linear. Show proof a. y(t) = 2x(t) – 5 b.·dy/dt + 4y(t) = 2x(t) C.dy/dt+2ty(t) = x(t) d.dy/dt+ 4y(t) + 1 = x(t) e.(dy/dt) + y(t) = x²(t) f. (dy/dt)² + y(t) = x(t)

8. Fort the electrical network below, Derive the two differential equations that relate input f(t) to that of outputs mesh current yı(t) and y₂(t). a. Now using substitution or Cramer's rule like we did in class find the differential equation that describes the relation between input f(t) and output yı(t) only.Similarly derive the differential equation that models the relations between input f(t) and output y2(t) only. b.Note that you are not required to solve the for the time domain algebraic expression of the two mesh currents. You are only required to set up the differential equations. We will see on how to solve them later on.

7. For the RLC circuit of your first HW, find the 2nd order differential equation that relates input-output (i.e., describes the system) if the output is the inductor voltage

1. Determine whether the following signals are energy signals, power signals or neither.Clearly show proof.

5. Which of the following systems are time-varying or time invariant. Show proof a. y(t) = x(t− 3) b.y(t) = x(t) cos(wt) C.d^2y/dt² + 2ty(t) = x(t)