Give an example of a 2 x 2 matrix A such that the linear system i = Ax satisfies all theconditions(a) has only one equilibrium.(b) is marginally stable but not asymptotically stable.(c) the solution x(t) is real for all initial conditions x(0) = [xo xo]T with ro E R.%3D

To give: An example of a 2×2 matrix A such that the linear system x˙=Ax satisfies all the following…

Answered: Give an example of a 2 x 2 matrix A…

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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Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 - 24e-1 -9e-21 060 -5 x'= 10 1x, x(0)= 8e3t x(t)= 1 1 0 -4e-t 3e-21 3t 4e -2 -20e-t 3e-21 3t 4e 060 (a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A = 10 1 validate the following identities and write the column vector that equals each side of the equation. 1 1 0 ×₁' = Ax₁ = x2' = Ax₂ = x3' = Ax3 (b) Next, compute the Wronskian of X(t). W[x()()()]= Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix. (c) Find x(t) = ☐ (d) x(t) = x(t)x¯1 (0)x₁ =
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Give an example of a 2 x 2 matrix A such that the linear system i = Ax satisfies all the conditions (a) has only one equilibrium. (b) is marginally stable but not asymptotically stable. (c) the solution x(t) is real for all initial conditions x(0) = [ro xo]T with ro E R.