5. (a) Given the system y[n] =(i)(ii)(iii) Causal(iv) Stable(v)(vi) LinearMemorylessInvertibleTime invariant(3x[n],(0,x[n] < 0x[n] ≥ 0determine whether this system is

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Answered: 5. (a) Given the system y[n] = (i) (ii)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the 3 first-order systems (1) x = 3x - 2y y = 4x - y x = 3x - 2y ý = 2x - 2y (3) x = -2x + y y = x - 2y (a) For each system identify the coefficient matrix and its eigenvalues. (b) Using the information from (a), classify each equilibrium at o for each system according to its type and stability. (c) Use Sage to plot a slope field for each system. (d) For system (3), find the solution to the IVP x(0) = 3, y(0) = 1.
7.(0) What is the Fisher linear discriminant method. (b) Given the 2-d data below for two classes: al-[(1,1), (1,2), (1,4), (2,1), (3,1), (3,3)], m2-[(2,2), (3,2), (3,4), (5,1), (5,4), (5.5)). Determine the optimal projection line in a single dimension.
A linear continuous time system is given by a state space model with A-[ ]), a-[1]. B = В C = [1 - 1], D=0 4 Analyze the following properties of the above system for different a: (i) asymptotic stability (ii) controllability, and (iii) observability.
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et Let 4(t) = ( be a fundamental matrix of the homogeneous system 2 1 x'(t) = ( { 2 ) ×(0). Which of the following is a particular solution to the nonhomogeneous system x10) = (; })«() + ( ) * 2 1 |x(t) + 4e ? Lütfen birini seçin: 2t +1 et :). 2t - 1 (1)* 2t -2t +1 -2t –1 2t – 1 2t +1 (*). 2t +2 et 2t – 2
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We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB) treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB infected individuals (including latent and infectious individuals) as well as the costs required to con- trol TB by treating latent and infectious individuals, over a certain time horizon [0,tf]. The cost of each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the optimization problem below. Minimize the objective function ' A₁E(t) + A₂I(t) + ¹¹³u²(t) + Bu²(t) J(u(.)), u(·) = "h 1 subject to dS = A(1p) BSI+ kV - ds, dt dV dt dE ུ|ཙ⪜ཊྛི|༴ཁྱི|ཙ dt dt = Ap-kV - dV, = BSI - (u(t) ++ d)E + OR, dR = E-(d--u2(t))I, = u₁(t)Eu(t) (d+ 0)R S(0) = So≥0, V(0) = √≥0, E(0) = Eo ≥0,1(0) = 10 ≥0, R(0) = Ro≥ 0. The control variables are assumed to be bounded: Given any t > 0, 0
Exercise in Class on 15th November 2022 A nonlinear function is given as f(x)= a sin x + bx² + In x.The function passes through three locations at x =1, x = and x = 2. The values of the function at the three locations are 2.1829, 6.0795 and 4.5117, respectively. (i) Organize the problem in a matrix form. (ii) Determine the value of coefficients a, b and c using Gauss elimination. (iii) Instead of using the elimination approach, use Gauss-Siedel iteration to solve for the value of coefficients a, b and c
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24 Use Use linearization (either with the Jacobian matrix or a change of variables defined by a translation....though you could do both =y=x² for practice!) to classify the equilibrium point at Y₁ = (x, y) = (1, 1) for the nonlinear system Yes, you dy =y-x dt can also determine/check the answer with StreamPlot on Mathematica, but I would encourage you to do it symbolically both ways mentioned above. Select one: O a. Yo = (x, y) = (1, 1) is a real (non-spiral) sink. O b. Yo = (x, y) = (1, 1) is a saddle point. O c. Yo = (x, y) = (1, 1) is a spiral source. O d. Yo = (x, y) = (1, 1) is a spiral sink. Check which theorem justifies the use of linearization (with a Jacobian matrix) to classify a hyperbolic equilibrium point Y₁ of a nonlinear system d = F(Y)? Select one: O a. Existence and Uniqueness Theorem O b. Poincaré-Bendixson Theorem O c. Spectral Theorem O d. Hartman-Grobman Theorem Check C An orthogonal matrix P has orthonormal columns and orthonormal rows. Select one: O True O False
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5. (a) Given the system y[n] = (i) (ii) (iii) Causal (iv) Stable (v) (vi) Linear Memoryless Invertible Time invariant (3x[n], (0, x[n] < 0 x[n] ≥ 0 determine whether this system is