91影视

Use the concept of a continuous dynamical system.Solve the differential equation $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{鈭�}\mathit{k}\mathit{x}$. Solvethe system$\frac{\mathbf{d}\stackrel{\mathbf{鈫�}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$ whenAis diagonalizable overR,and sketch the phase portrait for 2脳2 matricesA.

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

5.$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}\mathit{y}$with$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{8}$

Solve the differential equation$\mathbf{f}\mathbf{\text{'}}\left(t\right)\mathbf{-}\mathbf{2}\mathbf{f}\left(t\right)\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}$and find the solution of the differential equation.

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{f}\mathit{x}$as$\frac{\mathbf{d}\mathbf{x}}{\mathbf{f}\mathbf{x}}\mathbf{=}\mathit{d}\mathit{t}$and integrate both sides.

Find all complex solutions of the system.

$\frac{\mathbf{d}\stackrel{\mathbf{鈫�}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]\stackrel{\mathbf{鈫�}}{\mathbf{x}}$

In the form given in Theorem 9.2.3. What solution do you get if you let ${\mathit{c}}_{\mathbf{1}}\mathbf{=}{\mathit{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1}$?

Solve the differential equation$\mathbf{f}\mathbf{\text{'}}\mathbf{\text{'}}\left(t\right)\mathbf{+}\mathbf{f}\mathbf{\text{'}}\left(t\right)\mathbf{-}\mathbf{12}\mathbf{f}\left(t\right)\mathbf{=}\mathbf{0}$and find solution of the differential equation.

Determine the stability of the system

$\frac{\mathbf{d}\stackrel{鈫�}{x}}{\mathrm{dt}}\mathbf{=}\left[\begin{array}{cc}-1& 2\\ 3& -4\end{array}\right]\stackrel{鈫�}{x}$

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{f}\mathit{x}$ as$\frac{\mathbf{d}\mathbf{x}}{\mathbf{f}\mathbf{x}}\mathbf{=}\mathit{d}\mathit{t}$and integrate both sides.

Describe the behavior of your solution as t increases.

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

$\mathbf{T}\left(f\right)\mathbf{=}\mathbf{f}\mathbf{\text{'}}\mathbf{鈥�}\mathbf{f}\mathbf{}\mathbf{from}\mathbf{}\mathbf{C}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{}\mathbf{to}\mathbf{}\mathbf{C}\mathbf{\text{'}}\mathbf{\text{'}}$

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{f}\mathit{x}$as $\frac{\mathbf{d}\mathbf{x}}{\mathbf{f}\mathbf{x}}\mathbf{=}\mathit{d}\mathit{t}$and integrate both sides.

Consider a system$\frac{\mathbf{d}\stackrel{\mathbf{鈫�}}{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$where A is a symmetric matrix. When is the zero state a stable equilibrium solution? Give your answer in terms of the definiteness of the matrix A.