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In Exercises 3–6, solve the initial value problem for .
Chapter 5, Problem 3E(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.
\(A=\left[\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right]\)
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.
\(A=\left[\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right]\)
ANSWER:Solution 3EStep 1 The objective is to find the position of the particle at time , when the particle moving in a planar force field has the position vector that satisfies with the condition, .Consider the following matrix, The characteristic polynomial of the matrix : The Eigen values of the matrix are .