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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q7E (page 437)

Determine the stability of the system
$\frac{d \overset{鈫�}{x}}{dt} = [\begin{matrix} - 1 & 2 \\ 3 & - 4 \end{matrix}] \overset{鈫�}{x}$

Short Answer

Expert verified

Thus, the solution is unstable equilibrium.

Step by step solution

01

Given in the question.

The given system is:

$\frac{d \overset{鈫�}{x}}{d t} = [\begin{matrix} 鈭� 1 & 2 \\ 3 & 鈭� 4 \end{matrix}] \overset{鈫�}{x}$

02

Step 2:Solve for Eigen values.

The given determinant is solved as follows to obtain Eigen values.

$\begin{array}{l} \det ([\begin{matrix} 鈭� 1 & 2 \\ 3 & 鈭� 4 \end{matrix}] 鈭� [\begin{matrix} 位 & 0 \\ 0 & 位 \end{matrix}]) = 0 \\ \det ([\begin{matrix} 鈭� 1 鈭� 位 & 2 \\ 3 & 鈭� 4 鈭� 位 \end{matrix}]) = 0 \\ 位^{2} + 5 位鈭� 2 = 0 \\ 位 = \frac{鈭� 5 卤 \sqrt{25 鈭� 4 (1) (鈭� 2)}}{2} \\ 位 = \frac{鈭� 5 卤 \sqrt{33}}{2} \end{array}$

Thus, there are two real roots that is one positive and one negative. But both are not negative so, the solution is unstable equilibrium.

Hence, the solution is unstable equilibrium.

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Most popular questions from this chapter

Consider an $n 脳 n$ matrix A with m distinct eigenvalues $位_{1}, 位_{2}, 鈥�, 位_{m}$ .

(a) Show that the initial value problem $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ withrole="math" localid="1660807946554" $\overset{鈫�}{x} (0) = {\overset{鈫�}{x}}_{0}$ has a unique solutionrole="math" localid="1660807989045" $\overset{鈫�}{x} (t)$

(b) Show that the zero state is a stable equilibrium solution of the system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ if and only if the real part of all the $位_{i}$ is negative.Hint: Exercise 47 and Exercise 8.1.45 are helpful.

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

9. $\frac{d x}{d t} = x^{k} (w i t h 鈥� k 鈮� 1), x (0) = 1$

Consider a system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ where A is a $2 脳 2$ matrix with $tr A < 0$ . We are told that A has no real eigenvalue. What can you say about the stability of the system

Find the real solution of the system $\frac{d \overset{鈫�}{x}}{d t} = [\begin{matrix} 2 & 4 \\ - 4 & 2 \end{matrix}] \overset{鈫�}{x}$

Solve the differential equation $f''' (t) - f'' (t) - 4 f' (t) + 4 f (t) = 0$ and find all the real solution of the differential equation.

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