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2. (a) Let A be a square matrix of order tv 11 with the corresponding and V = Find exponential solutions (t) = = equations x'' = AX.

2. (a) Let A be a square matrix of order tv 11 with the corresponding and V = Find exponential solutions (t) = = equations x'' = AX.

Advanced Engineering Mathematics
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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2. (a) Let **A** be a square matrix of order two which has eigenvectors \(\vec{u} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}\) and \(\vec{v} = \begin{bmatrix} 4 \\ 1 \end{bmatrix}\) with corresponding eigenvalues \(\lambda = 2\) and \(\lambda = -5\). Find exponential solutions \(\vec{X}(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}\) of the system of differential equations \(\vec{X}' = \mathbf{A} \vec{X}\). (b) How do \(x_1(t)\) and \(x_2(t)\) behave in the found solutions as \(t \to \infty\)?
Transcribed Image Text:2. (a) Let **A** be a square matrix of order two which has eigenvectors \(\vec{u} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}\) and \(\vec{v} = \begin{bmatrix} 4 \\ 1 \end{bmatrix}\) with corresponding eigenvalues \(\lambda = 2\) and \(\lambda = -5\). Find exponential solutions \(\vec{X}(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}\) of the system of differential equations \(\vec{X}' = \mathbf{A} \vec{X}\). (b) How do \(x_1(t)\) and \(x_2(t)\) behave in the found solutions as \(t \to \infty\)?
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