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

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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.
4. Let (*) be the following homogeneous system F: of (n-1) linear equations in n unknows All X₁ + A₁₂ X₂ t + An Xn = 0. a22X2 A21X₁ + a₂₂X₂ + ... + Cl₂n Xn = 0. An-1,1X₁+ an-1, 2X₂ + + On-in Xn = 0. ,2 Is Suppose that no equation in this system linear combination of the other equations. For each =1,2,.nlet Ai & Ma-x) (F) be the matrix obtained by deleting the ith column of the coefficient matrix of (*), and let det (Ai). Ci over it! = (*) (-1) Prove the following statements (a) Not all of C₁,..., Cn are 0. (5) The solution space of the system (*) is olle-dimen (C) A vector (X₁. --, Xn) € F" is a -sional, solution to the system (*) iff (X₁...., Xn) = t (C₁, , (n) a ***/ for some te F.
consider the nonhomogeneous linear system { dr # = a1 (t)x + bị1(t)y + fi(t). # = a2 (t)x + b2(t)y + f2(t). (*) and the corresponding homogeneous system { dr * = a1 (t)x + b1(t)y. fip * = a2(t)x + b2(t)y. || (**) a) Suppose that Sr = x1(t) ly = y1(t). x = r2(t) ly = y2(t). and are linearly independent solutions of equation(**).so, that x = c1x1(t) + c2r2(t) y = c1y1(t) + c2y2(t). || is its general solution. Then show that x = vi(t)x1(t) + v2r2(t) y = v1(t)y1(t) + v2(t)y2(t). is a particular solution of Equation(*) if the function v1,v2 satisfy the system víy1+v2¥2 (t)=f2. this technique for finding particular solutions of nonhomogeneous linear system is called the method of variation of parameters. b) Apply the method described in part (a) to find particular solution of the nonhomogeneous system IP = x + y – 5t +2. dt 3D 4х — 4.x 2у — 8t — 8.
(5) Consider the family of systems 4 = (" Y (a) Identify all bifurcation values for this family. (b) Describe the bifurcations that occur as m increases.
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5. Find your equivalent system of the equation and find the region of stability of the equation x" + ax' + (1 – a)x = 0
4. (S.10). Use Gaussian elimination with backward substitution to solve the following linear system: 2.r1 + 12 – 13 = 5, 1 + 12 – 3r3 = -9, -I1 + 12 +2r3 = 9;
Find a particular solution of the indicated linear system that satisfies the initial conditions x₁ (0) = 5, x₂ (0) = 5, and x3 (0) = 8. 3 [1-4-1-4 3t x' = 12 16 1 x; x₁ =e -2, X₂= 2 - 15 - 19 - 12 - 12 3 The particular solution is x₁ (t) = X₂ (t) = and x3 (t): = 4t x3 = e 0
1. Use Theorem 1 to solve the nonhomogeneous linear system *-[-])*+(1) x(0) = (6). with the initial condition =
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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