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Answer: In Exercises 3–6, solve the initial value problem
Chapter 5, Problem 5E(choose chapter or problem)
In Exercises 3–6, solve the initial value problem \(\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)\) for \(t \geq 0\), with \(\mathbf{x}(0)=(3,2)\). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by \(\mathbf{x}^{\prime}=A \mathbf{x}\). Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.
\(A=\left[\begin{array}{rr}7 & -1 \\ 3 & 3\end{array}\right]\)
Questions & Answers
QUESTION:
In Exercises 3–6, solve the initial value problem \(\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)\) for \(t \geq 0\), with \(\mathbf{x}(0)=(3,2)\). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by \(\mathbf{x}^{\prime}=A \mathbf{x}\). Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.
\(A=\left[\begin{array}{rr}7 & -1 \\ 3 & 3\end{array}\right]\)
ANSWER:Solution 5EStep 1 Consider the following dynamic system with initial condition: and Here, .The characteristic polynomial of the matrix : The Eigen values of the matrix are .