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Chapter 9: Q33E (page 426)

Sketch rough phase portraits for the dynamical systems given in Exercises 32 through 39.

33.

dx鈫�dt=[-432-3]x鈫�

Short Answer

Expert verified

Sketch the lines that determent by eigenvectors and depend on the signs of Eigen values sketch the arrows that depict the future and past of the system.

Step by step solution

01

Determine the Eigen values and Eigen vectors

Consider that both 位1鈮�0 and 位2鈮�0. When位i corresponding to eigenvectorv鈫�i would be greater than 0, then the arrows on the line determent by vector v鈫�ipoint away from the origin and when 位i<0then they point towards to the origin.

02

Sketch the rough phase portraits for the dynamical systems

There are three cases: 位1<位2<0,位1>位2>0and 位1<0<位2.

When 位i=0trajectories are rays parallel to v鈫�jand arrows on those trajectories converge into the line determent by v鈫�j when 位j<0and they diverge from it when 位j>0 .

Sketch the rough phase portraits for the dynamical system as follows:

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Most popular questions from this chapter

Ngozi opens a bank account with an initial balance of 1,000 Nigerian naira. Let b(t) be the balance in the account at time t; we are told that b(0) = 1,000. The bank is paying interest at a continuous rate of 5% per year. Ngozi makes deposits into the account at a continuous rate of s(t) (measured in naira per year). We are told that s(0) = 1,000 and that s(t) is increasing at a continuous rate of 7% per year. (Ngozi can save more as her income goes up over time.)

(a) set up a linear system of the form|dbdt=?b+?sdsdt=?b+?s|

(b) Find b(t)ands(t).

Consider the IVP dx鈫�dt=Ax鈫�withx鈫�(0)=x鈫�0where A is an upper triangularn脳nmatrix with m distinct diagonal entries 位1,..鈥�,位m. See the examples in Exercise 45 and 46.

(a) Show that this problem has a unique solutionx鈫�(t)whose componentsxi(t)are of the form

xi(t)=P1(t)e位1t+...+Pm(t)e位mt,

for some polynomials Pj(t).Hint: Find first xn(t), then xn-1(t), and so on.

(b) Show that the zero state is a stable equilibrium solution of this system if (and only if) the real part of all the位i is negative.

Solve the systemdx鈫�dt=[710-4-5]x鈫�withx鈫�(0)=[10]. Give the solution in real form. Sketch the solution.

Use the concept of a continuous dynamical system.Solve the differential equation dxdt=鈭�kx. Solvethe system dx鈫�dt=Ax鈫�when Ais 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.

2. dxdt=-0.71xwith x(0)=-e.

Solve the systemdx鈫�dt=[-11-21]x鈫�withx鈫�(0)=[01]. Give the solution in real form. Sketch the solution.
See all solutions

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