-3Fundamental matrix associated with the homogeneous system x' =2(a) Φ(t)=|(b) Φ(t)=|(c) Φ(t)=|(d) Þ(t)=|-21e-21-Ste-2e-St-51e-Ste-51-3e-2t 5e-Ste-21-3e-2t-5e-St1-4x is,

We have to find fundamental matrix.

Answered: -3 1 Fundamental matrix associated with…

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 system x0 = Ax for A = 1 3 0 2 . Find eAt . (That is, fifind a fundamental matrix Φ(t) with Φ(0) = I).
dx dt = x+y+z; (a) (b) dy dt = x + 3y - z; dz dt = 2y + 2z For the system above, define matrix A and vector X and write the system in matrix form. Show all work to solve the system and clearly state the general solution vector X
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 =
2. (i) Let A be an invertible matrix. Assume that A = A-1, What are the possible values for det(A)? (ii) Let B be a skew-symmetric matrix over R, i.e. AT = -B and of order 2n-1. Show that det(B) = 0. (iii) Show that if a square matrix C is hermitian, i.e. C* = C then det(C) is a %3D real number.
Verify that X(t) is a fundamental matrix for the given system and compute X ¹(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t)= X(t)X¹(0)xo is the solution to the initial value problem x'=Ax, x(0)=xo- 060 x = 1 0 1 x, x(0) = 110 1 equals each side of the equation. x₁' = Ax₁ = x₂ = Ax₂ = X3' = Ax3 = -3 X(t) = 060 (a) If X(t) = [X₁ (t) x₂ (t) x3 (t)] and A= 1 0 1 110 6e-t-3e-2t 8e³t e-2t 4631 e-2t 4e3t -e-t -5e-t validate the following identities and write the column vector that
3. Consider the system 3 -2 %3D -2 (a) Find a fundamental matrix V(t). (b) Find the special fundamental matrix (t) so that (0) = I. (c) Based on the result in part (b), show that D(t)Ð(s) = P(t+s) by multiplying the matrices (t) and (s).

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-3 1 Fundamental matrix associated with the homogeneous system x' = 2 -4 (a) (t)= 9 (b) (t)= (c) (t)= ev (d) Φ(t)= -2e-St -2e-2t -3e-2r -21 -3e-21 Se-St -Se-St x is,