91Ó°ÊÓ

Question:Consider the system

$\frac{\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left[\begin{array}{cc}{\mathrm{λ}}_1& 0\\ 0& {\mathrm{λ}}_2\end{array}\right]\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{.}$

For the values of and given in Exercises 16 through 19, sketch the trajectories for all nine initial values shown in the following figure. For each of the points, trace out both the future and the past of the system.

$\mathbf{17}\mathbf{.}\mathbf{}{\mathit{λ}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{λ}}_{\mathbf{2}}\mathbf{=}\mathbf{2}$

Solve the differential equation$\mathbf{f}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{2}\mathbf{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{sin}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$and find all the real solutions of the differential equation.

Consider the system $\frac{\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left(\begin{array}{cc}-1& K\\ K& -1\end{array}\right)\stackrel{\mathbf{→}}{\mathbf{x}}$where k is an arbitrary constant. For which values of k is the zero state a stable equilibrium solution?

Solve the differential equation$\mathbf{f}\mathbf{"}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{3}\mathbf{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{2}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{cos}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$and find all the real solutions of the differential equation.

For the values of ${\mathit{λ}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{1}$and ${\mathit{λ}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{2}$, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.

Consider a diagonalizable$\mathbf{3}\mathbf{×}\mathbf{3}$matrix A such that the zero state is a stable equilibrium solution of the system$\frac{\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$. What can you sayabout the determinant and the trace of A.

True or False? If the trace and the determinant of a $\mathbf{3}\mathbf{×}\mathbf{3}$matrix A are both negative, then the origin is a stable equilibrium solution of the system$\frac{\mathbf{d}\stackrel{\mathbf{→}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$. Justify your answer.

Solve the differential equation$\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{{\mathbf{dt}}^{\mathbf{2}}}\mathbf{+}\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{cos}\left(t\right)$and find all the real solutions of the differential equation.

For the values of ${\mathit{λ}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$and ${\mathit{λ}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.

Use the concept of a continuous dynamical system.Solve the differential equation $\frac{dx}{dt}=−kx$. Solvethe system$\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$ whenAis diagonalizable overR,and sketch the phase portrait for 2×2 matricesA.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.