(a) Solve the initial value problem (d 𝐮)/(d t)=( -1 2 -1 -3 ) 𝐮, 𝐮(0)=( 1

step 1 So we have this one. It's still going to be a separation of variables, right? So this is another

(a) Solve the initial value problem (d 𝐮)/(d t)=( -1 ...

(a) Solve the initial value problem $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}-1 & 2 \\ -1 & -3\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$. (b) Sketch a picture of your solution curve $\mathbf{u}(t)$, indicating the direction of motion. (c) Is the origin (i) stable? (ii) asymptotically stable? (iii) unstable? (iv) none of these? Justify your answer.

(a) Solve the initial value problem $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}-1 & 2 \\ -1 & -3\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$.
(b) Sketch a picture of your solution curve $\mathbf{u}(t)$, indicating the direction of motion.
(c) Is the origin (i) stable? (ii) asymptotically stable? (iii) unstable? (iv) none of these? Justify your answer.

Key Concepts

Linear Systems of Differential Equations

This concept involves studying a system of equations where the rate of change of the variable vector is given by a constant coefficient matrix multiplying the vector. The method of solving these systems often involves techniques that exploit linear algebra, which can transform the system into decoupled equations using properties like eigenvalues and eigenvectors.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental in analyzing linear systems because they provide insight into the behavior of the system over time. They allow the original system to be decomposed into simpler, independent modes of motion, and are crucial in determining properties such as stability and the qualitative nature of the solution trajectories.

Matrix Exponential

The matrix exponential is a tool used to solve linear systems of differential equations with constant coefficients. By exponentiating the coefficient matrix, one obtains a fundamental matrix solution which, when applied to an initial condition, yields the solution of the system for all times.

Phase Portrait

The phase portrait is a graphical representation of the trajectories of a dynamical system in the state-space. It provides insight into the behavior of solutions, indicating directions, equilibrium points, and qualitative features like spirals, nodes, and saddles, without necessarily determining precise solution values.

Stability Analysis

Stability analysis involves determining whether small perturbations in the initial conditions lead to solutions that remain close to a particular point, such as an equilibrium, or diverge away over time. It distinguishes between types of stability, including stability in the sense of Lyapunov and asymptotic stability, by examining the eigenvalues of the system matrix.

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