Q16E Consider the system dx鈫抎t=(01a... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q16E (page 439)

Consider the system $\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) \overset{鈫�}{x}$ where a and b are arbitrary constants for which values of a and b is the zero state a stable equilibrium solution?

Short Answer

Expert verified

The value of arbitrary constants a and b in the zero state a stable equilibrium solution is $a = 1$ and $b = - 1$

Step by step solution

Explanation of the stability of a continuous dynamical system

For a system, $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ here A is the matrix form.

The zero state is an asymptotically stable equilibrium solution if and only if the real parts of all eigen values of A are negative.

Step 2:Solution for the values of k is the zero state a stable equilibrium solution

Consider the system $\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) \overset{鈫�}{x}$ where a and b are arbitrary constants

Here $A = (\begin{matrix} 0 & 1 \\ a & b \end{matrix})$ which is the real $2 脳 2$ matrix.

$\begin{array}{rcl} A - 位 I & = & 0 \\ (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) - 位 (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) & = & 0 \\ (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) - (\begin{matrix} 位 & 0 \\ 0 & 位 \end{matrix}) & = & 0 \\ (\begin{matrix} - 位 & 1 \\ a & b - 位 \end{matrix}) & = & 0 \\ (- 位) (b - 位) - a & = & 0 \\ - 位 b + 位^{2} - a & = & 0 \\ 位^{2} - b 位 - a & = & 0 \end{array}$

Here $位$ is zero state according to the stability equilibrium condition because it is zero state in the asymptotically stable equilibrium.

So value for $b = - 1$ and $a = 1$ and it is obtained from the zero state condition of asymptotically stable equilibrium

Thus, the value for the zero statesof a stable equilibrium solution is $a = 1$ and $b = - 1$ .

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91影视

Consider the system $\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) \overset{鈫�}{x}$ where a and b are arbitrary constants for which values of a and b is the zero state a stable equilibrium solution?

Short Answer

Step by step solution

Explanation of the stability of a continuous dynamical system

Step 2:Solution for the values of k is the zero state a stable equilibrium solution

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91影视

Consider the system dx鈫�dt=(01ab)x鈫�where a and b are arbitrary constants for which values of a and b is the zero state a stable equilibrium solution?

Short Answer

Step by step solution

Explanation of the stability of a continuous dynamical system

Step 2:Solution for the values of k is the zero state a stable equilibrium solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider the system $\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) \overset{鈫�}{x}$ where a and b are arbitrary constants for which values of a and b is the zero state a stable equilibrium solution?