Chapter 9.9, Problem 1TFR

True-False Review

For Questions (a)-(f), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) An equilibrium point is a point toward which all trajectories of a linear system of differential equations approach as $t\to \infty$.

(b) A linear system $x^{\prime }=Ax$ for which $A$ has complex eigenvalues $\lambda =\pm ib$ (for some $b\in ℝ$) gives rise to elliptical trajectories in the phase plane.

(c) The equilibrium point of a linear system $x^{\prime }=Ax$ that has two positive (real) eigenvalues is called an unstable node.

(d) If all solutions to the linear system $x^{\prime }=Ax$ approach $\left(0,0\right)$, then $\left(0,0\right)$ is stable.

(e) The type of equilibrium point for the linear system $x^{\prime }=Ax$ is the same as the type of equilibrium point for the linear system $x^{\prime }=\left(2A\right)x$.

(f) For a saddle point, the only trajectories that approach $\left(0,0\right)$ as $t\to \infty$ are those pointing along the eigenvector $v$ corresponding to the positive eigenvalue $\lambda$.