Consider the system with

Chapter 9, Problem 17E

(choose chapter or problem)

17. Consider the system with \(x^{\prime}(t)=A x(t), t \geq 0\), with

\(A=\left[\begin{array}{cc}  1 & \sqrt{3} \\  \sqrt{3} & -1  \end{array}\right]\)

(a) Show that the matrix \(A\) has eigenvalues \(r_{1}=2 \text { and } r_{2}=-2\) with corresponding eigenvectors \(u_{1}=\operatorname{col}(\sqrt{3}, 1)\)  and \(u_{2}=\operatorname{col}(1,-\sqrt{3})\).

(b) Sketch the trajectory of the solution having initial vector \(x(0)=-u_{1}\).

(c) Sketch the trajectory of the solution having initial vector \(x(0)=-u_{2}\).

(d) Sketch the trajectory of the solution having initial vector \(x(0)=u_{2}-u_{1}\).

Equation Transcription:

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Text Transcription:

x'(t)=Ax(t), t \geq 0

A=[  \sqrt 3    -1    1   \sqrt 3]

A      

r1=2 and r2=-2      

u1=col (\sqrt 3, 1)      

u2=col (1,-\sqrt 3)    

x(0)=-u1

x(0)=-u2

x(0)=u2-u1