Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Find the real solution of the system $\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$

The solution is $\vec{x} (t) = [\begin{matrix} \cos (3 t) & - \sin (3 t) \\ \sin (3 t) & \cos (3 t) \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: definition of the theorem.

Continuous dynamical system with eigenvalues

Consider the linear system $\frac{d \vec{x}}{d t} = A \vec{x}$ where A is a real $2 \times 2$ matrix with complex eigenvalues role="math" localid="1659881702011" $p \pm i q$ (and $q \neq 0$ ).

Consider an eigenvector $\vec{v} + i \vec{w}$ with eigenvalue $p \pm i q$ .

Then $\vec{x} (t) = e^{p t} S [\begin{matrix} c o s (q t) & - s i n (q t) \\ s i n (q t) & c o s (q t) \end{matrix}] S^{- 1} x_{0}$ where $S = [\vec{w} \vec{v}]$ . Recall that $S^{- 1} x_{0}$ is the coordinate vector of ${\vec{x}}_{0}$ with respect to basis $\vec{w}, \vec{v}$ .

Step2: To find the eigenvalues

Consider the given system as follows.

$\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$

No, to find the eigenvalues of the coefficient matrix as follows.

$|\begin{matrix} 0 - λ & - 3 \\ 3 & 0 - λ \end{matrix}| = 0 (0 - λ) (0 - λ) - (3) (- 3) = 0 λ^{2} + 9 = 0 λ^{2} = - 9$

Simplify further as follows

$λ^{2} = - 9 λ = \pm 3 i$

Therefore, the eigenvalues are $λ = 3 i a n d λ = - 3 i$

Step3: To find the eigenvectors

To find a formula for trajectory as follows.

$E_{3 i} = k e r [\begin{matrix} 0 - 3 i & - 3 \\ 3 & 0 - 3 i \end{matrix}] = s p a n ([\begin{matrix} 0 \\ 1 \end{matrix}] + i [\begin{matrix} 1 \\ 0 \end{matrix}])$

Therefore, the eigenvectors are as follows.

$\vec{v} = [\begin{matrix} 0 \\ 1 \end{matrix}] \vec{w} = [\begin{matrix} 1 \\ 0 \end{matrix}]$

Then by the theorem, the general solution for the system $\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$ is as follows.

$\vec{x} (t) = e^{p t} S [\begin{matrix} c o s (q t) & - s i n (q t) \\ s i n (q t) & c o s (q t) \end{matrix}] S^{- 1} {\vec{x}}_{0}$ where $S = [\vec{w} \vec{v}]$

Similarly, the values of p,q and S are as follows.

$p = 0 q = 3$

$S = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Substitute the value 0 for p and 3 for q and $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ for S in $\vec{x} (t) = e^{p t} S [\begin{matrix} c o s (q t) & - s i n (q t) \\ s i n (q t) & c o s (q t) \end{matrix}] S^{- 1} {\vec{x}}_{0}$ as follows.

$\vec{x} (t) = e^{p t} S [\begin{matrix} c o s (q t) & - s i n (q t) \\ s i n (q t) & c o s (q t) \end{matrix}] S^{- 1} {\vec{x}}_{0} \vec{x} (t) = e^{0 t} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} c o s (3 t) & - s i n (3 t) \\ s i n (3 t) & c o s (3 t) \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] \vec{x} (t) = [\begin{matrix} 1 (c o s (3 t)) + 0 (s i n (3 t)) & 1 (- s i n (3 t)) + 0 (c o s (3 t)) \\ 0 (c o s (3 t)) + 1 (s i n (3 t)) & 0 (- s i n (3 t)) + 1 (c o s (3 t)) \end{matrix}] \vec{x} (t) = [\begin{matrix} c o s (3 t) & - s i n (3 t) \\ s i n (3 t) & c o s (3 t) \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

Hence, the solution for the system $\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$ is $\vec{x} (t) = [\begin{matrix} \cos (3 t) & - \sin (3 t) \\ \sin (3 t) & \cos (3 t) \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

where x and y are arbitrary constants.

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the real solution of the system $\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$

Step by Step Solution

Step1: definition of the theorem.

Step2: To find the eigenvalues

Step3: To find the eigenvectors

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the real solution of the system dx→dt=[0-330]x→

Step by Step Solution

Step1: definition of the theorem.

Step2: To find the eigenvalues

Step3: To find the eigenvectors

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Find the real solution of the system $\frac{d \vec{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \vec{x}$