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Magazine Problem 5
Use Theorem \(7.4\) to classify the equilibrium point at the origin as unstable, stable, or asymptotically stable. Verify your findings by sketching a phase portrait of the system. \(y^{\prime}=\left(\begin{array}{rr}0.1 & 2.0 \\ -2.0 & 0.1\end{array}\right) y\)
Problem 5
The matrix \(A\) has real eigenvalues. Find the general solution of the system \(\mathbf{y}^{\prime}=A \mathbf{y}\). \(A=\left(\begin{array}{rr}1 & 2 \\ -1 & 4\end{array}\right)\)
Problem 5
Classify the equilibrium point of the system \(\mathbf{y}^{\prime}=A \mathbf{y}\) based on the position of \((T, D)\) in the trace-determinant plane. Sketch the phase portrait by hand. Verify your result by creating a phase portrait with your numerical solver. \(A=\left(\begin{array}{rr}-11 & -5 \\ 10 & 4\end{array}\right)\)
Problem 5
Use hand calculations to find the characteristic polynomial and eigenvalues for each of the matrices. \(A=\left(\begin{array}{rr}5 & 3 \\ -6 & -4\end{array}\right)\)
Problem 5
Proposition \(5.1\) guarantees that eigenvectors associated with distinct eigenvalues are linearly independent. Each matrix has distinct eigenvalues; find them and their associated eigenvectors. Verify that the eigenvectors are linearly independent. \(A=\left(\begin{array}{rrr}-4 & 0 & 2 \\ 12 & 2 & -6 \\ -6 & 0 & 3\end{array}\right)\)
Problem 6
Classify the equilibrium point of the system \(\mathbf{y}^{\prime}=A \mathbf{y}\) based on the position of \((T, D)\) in the trace-determinant plane. Sketch the phase portrait by hand. Verify your result by creating a phase portrait with your numerical solver. \(A=\left(\begin{array}{rr}6 & -5 \\ 10 & -4\end{array}\right)\)
Problem 6
Use hand calculations to find the characteristic polynomial and eigenvalues for each of the matrices. \(A=\left(\begin{array}{rr}-2 & 5 \\ 0 & 2\end{array}\right)\)
Problem 6
The matrix \(A\) has real eigenvalues. Find the general solution of the system \(\mathbf{y}^{\prime}=A \mathbf{y}\). \(A=\left(\begin{array}{rr}-1 & 1 \\ 1 & -1\end{array}\right)\)
Problem 7
Find the general solution of the systems. \(x^{\prime}=4 x-5 y+4 z\) \(y^{\prime}=-y+4 z\) \(z^{\prime}=z\)
Problem 7
Use hand calculations to find the characteristic polynomial and eigenvalues for each of the matrices. \(A=\left(\begin{array}{rr}-3 & 0 \\ 0 & -3\end{array}\right)\)
