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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 10

Dynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

Choose one or more of the following differential equations, and then: (a) Solve the equation directly. (b) Write down its phase plane equivalent, and the general solution to the phase plane system. (c) Plot at least four representative trajectories to illustrate the phase portrait. $(d)$ Choose two trajectories in your phase portrait and graph the corresponding solution curves $u(t)$. Explain in your own words how the orbit and the solution graph are related. (i) $\ddot{u}+4 u=0$, (ii) $\ddot{u}-4 u=0$, (iii) $\ddot{u}+2 \dot{u}+u=0$, (iv) $\ddot{u}+4 \dot{u}+3 u=0$, (v) $\ddot{i}-2 \dot{u}+10 u=0$.

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Problem 1

Classify the following systems according to whether the origin is (i) asymptotically stable, (ii) stable, or (iii) unstable: (a) $\frac{d u}{d t}=-2 u-v, \frac{d v}{d t}=u-2 v$;
(b) $\frac{d u}{d t}=2 u-5 v$, $\frac{d v}{d t}=u-v$
(c) $\frac{d u}{d t}=-u-2 v, \frac{d v}{d t}=2 u-5 v ;$
(d) $\frac{d u}{d t}=-2 v, \frac{d v}{d t}=8 u$;
(e) $\frac{d u}{d t}=-2 u-v+w, \frac{d v}{d t}=-u-2 v+w, \frac{d w}{d t}=-3 u-3 v+2 w$;
(f) $\frac{d u}{d t}=-u-2 v, \frac{d v}{d t}=6 u+3 v-4 w, \frac{d w}{d t}=4 u-3 w$;
(g) $\frac{d u}{d t}=2 u-v+3 w, \frac{d v}{d t}=u-v+w ; \frac{d w}{d t}=-4 u+v-5 w$;
(h) $\frac{d u}{d t}=u+v-w, \frac{d v}{d t}=-2 u-3 v+3 w, \frac{d w}{d t}=-v+w$.

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Problem 1

For each the following: (a) Write the system as $\dot{\mathbf{u}}=A \mathbf{u}$. (b) Find the eigenvalues and eigenvectors of A. (c) Find the general real solution of the system. (d) Draw the phase portrait, indicating its type and stability properties: $(i) \dot{u}_1=-u_2, \dot{u}_2=9 u_1$,
(ii) $\dot{u}_1=2 u_1-3 u_2, \dot{u}_2=u_1-u_2$,
(iii) $\dot{u}_1=3 u_1-2 u_2, \dot{u}_2=2 u_1-2 u_2$.

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Problem 1

Find the exponentials $e^{t A}$ of the following $2 \times 2$ matrices:
(a) $\left(\begin{array}{ll}2 & -1 \\ 4 & -3\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$
(c) $\left(\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right)$
(d) $\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$,
(e) $\left(\begin{array}{ll}-1 & 2 \\ -5 & 5\end{array}\right)$
(f) $\left(\begin{array}{rr}1 & 2 \\ -2 & -1\end{array}\right)$

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01:44

Problem 1

A 6 kilogram mass is connected to a spring with stiffness $21 \mathrm{~kg} / \mathrm{sec}^2$. Determine the frequency of vibration in hertz (cycles per second).

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:24

Problem 1

Graph the following functions. Describe the fast oscillatory and beat frequencies:
(a) $\cos 8 t-\cos 9 t$, (b) $\cos 26 t-\cos 24 t$, (c) $\cos 10 t+\cos 9.5 t$, (d) $\cos 5 t-\sin 5.2 t$.

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 2

(a) Convert the third order equation $\frac{d^3 u}{d t^3}+3 \frac{d^2 u}{d t^2}+4 \frac{d u}{d t}+12 u=0$ into a first order system in three variables by setting $u_1=u, u_2=\dot{u}, u_3=\ddot{u}$. (b) Solve the equation directly, and then use this to write down the general solution to your first order system. (c) What is the dimension of the solution space?

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01:52

Problem 2

Write out the formula for the general real solution to the system in Example 10.17 and verify its stability.

Nick Johnson
Nick Johnson
Numerade Educator
02:49

Problem 2

For each of the following systems
(i) $\mathbf{\mathbf { u }}=\left(\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right) \mathbf{u}$,
(ii) $\dot{\mathbf{u}}=\left(\begin{array}{ll}1 & -1 \\ 5 & -3\end{array}\right) \mathbf{u}$,
(iii) $\mathbf{\mathbf { u }}=\left(\begin{array}{cc}-3 & 5 / 2 \\ -5 / 2 & 2\end{array}\right) \mathbf{u}:$
(a) Find the general real solution. (b) Using the solution formulas obtained in part (a), plot several trajectories of each system. On your graphs, identify the eigenlines (if relevant), and the direction of increasing $t$ on the trajectories. (c) Write down the type and stability properties of the system.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 2

Determine the matrix exponential $e^{t . A}$ for the following matrices:
(a) $\left(\begin{array}{rrr}0 & 0 & 0 \\ 2 & 0 & 1 \\ 0 & -1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$,
(c) $\left(\begin{array}{rrr}-1 & 1 & 1 \\ -2 & -2 & -2 \\ 1 & -1 & -1\end{array}\right)$,
(d) $\left(\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right)$.

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04:57

Problem 2

The lowest audible frequency is about 20 hertz $=20$ cycles per second. How small a mass would need to be connected to a unit spring to produce a fast enough vibration to be audible? (As always, we assume the spring has negligible mass, which is probably not so reasonable in this situation.)

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:47

Problem 2

Solve the following initial value problems: (a) $\ddot{u}+36 u=\cos 3 t, u(0)=0, \dot{u}(0)=0$.
(b) $\ddot{u}+6 \dot{u}+9 u=\cos t, u(0)=0, \dot{u}(0)=1$. (c) $\ddot{u}+\dot{u}+4 u=\cos 2 t, u(0)=1$,
$\dot{u}(0)=-1 .(d) \ddot{u}+9 u=3 \sin 3 t, u(0)=1, \dot{u}(0)=-1$. (e) $2 \ddot{u}+3 \dot{u}+u=\cos \frac{1}{2} t$,
$u(0)=3, \dot{u}(0)=-2$. (f) $3 \ddot{u}+4 \dot{u}+u=\cos t, u(0)=0, \dot{u}(0)=0$.

Andrija Isakov
Andrija Isakov
Numerade Educator
01:27

Problem 3

Convert the second order coupled system of ordinary differential equations
$$
\ddot{u}=a \dot{u}+b \dot{v}+c u+d v, \quad \ddot{v}=p \dot{u}+q \dot{v}+r u+s v,
$$
into a first order system involving four variables.

Christian Otero
Christian Otero
Numerade Educator
04:24

Problem 3

Write out and solve the gradient flow system corresponding to the following quadratic forms: (a) $u^2+v^2$, (b) $u v$, (c) $4 u^2-2 u v+v^2$, (d) $2 u^2-u v-2 u w+2 v^2-v w+2 w^2$.

Faizanullah Kazmi
Faizanullah Kazmi
Numerade Educator
02:28

Problem 3

Classify the following systems, and sketch their phase portraits.
(a) $\frac{d u}{d t}=-u+4 v$,
(b) $\frac{d u}{d t}=-2 u+v$,
(c) $\frac{d u}{d t}=5 u+4 v$,
(d) $\frac{d u}{d t}=-3 u-2 v$,
$\frac{d v}{d t}=u-2 v$.
$$
\frac{d v}{d t}=u-4 v \text {, }
$$
$$
\frac{d v}{d t}=u+2 v . \quad \frac{d v}{d t}=3 u+2 v .
$$

Nick Johnson
Nick Johnson
Numerade Educator
00:53

Problem 3

Verify the determinant formula of Lemma 10.28 for the matrices in Exercises 10.4.1 and 10.4.2.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
04:38

Problem 3

Graph the following functions. Which are periodic? quasi-periodic? If periodic, what is the (minimal) period? (a) $\sin 4 t+\cos 6 t$, (b) $1+\sin \pi t$, (c) $\cos \frac{1}{2} \pi t+\cos \frac{1}{3} \pi t$, (d) $\cos t+\cos \pi t$, (e) $\sin \frac{1}{4} t+\sin \frac{1}{5} t+\sin \frac{1}{6} t$, (f) $\cos t+\cos \sqrt{2} t+\cos 2 t$, (g) $\sin t \sin 3 t$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
00:40

Problem 3

Solve the following initial value problems. In each case, graph the solution and explain what type of motion is represented. (a) $\ddot{u}+4 \dot{u}+40 u=125 \cos 5 t, u(0)=0, \dot{u}(0)=0$,
(b) $\ddot{u}+25 u=3 \cos 4 t, u(0)=1, \dot{u}(0)=1$, (c) $\ddot{u}+16 u=\sin 4 t, u(0)=0, \dot{u}(0)=0$,
(d) $\ddot{u}+6 \dot{u}+5 u=25 \sin 5 t, u(0)=4, \dot{u}(0)=2$.

Ernest Castorena
Ernest Castorena
Numerade Educator
01:45

Problem 4

(a) Show that if $\mathbf{u}(t)$ solves $\mathbf{u}=A \mathbf{u}$, then its time reversal, defined as $\mathbf{v}(t)=\mathbf{u}(-t)$, solves $\dot{\mathrm{v}}=B \mathrm{v}$, where $B=-A$. (b) Explain why the two systems have the same phase portraits, but the direction of motion along the trajectories is reversed. (c) Apply time reversal to the system(s) you derived in Exercise 10.1.1. (d) What is the effect of time reversal on the original second order equation?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 4

Write out and solve the Hamiltonian systems corresponding to the first three quadratic forms in Exercise 10.2.3. Which of them are stable?

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02:04

Problem 4

Justify the solution formulas (10.32) and (10.33).

Chai Santi
Chai Santi
Numerade Educator
08:37

Problem 4

Solve the indicated initial value problems by first exponentiating the coefficient matrix and then applying formula (10.42):
(a) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right) \mathbf{u}, \quad \mathbf{u}(0)=\left(\begin{array}{r}1 \\ -2\end{array}\right)$,
(b) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{ll}3 & -6 \\ 4 & -7\end{array}\right) \mathbf{u}, \quad \mathbf{u}(0)=\left(\begin{array}{r}-1 \\ 1\end{array}\right)$,
(c) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}-9 & -6 & 6 \\ 8 & 5 & -6 \\ -2 & 1 & 3\end{array}\right) \mathbf{u}, \quad \mathbf{u}(0)=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$.

Ashley Boni
Ashley Boni
Numerade Educator

Problem 4

What is the minimal period of a function of the form $\cos \frac{p}{q} t+\cos \frac{r}{s} t$, assuming that each fraction is in lowest terms, i.e., its numerator and denominator have no common factors?

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04:35

Problem 4

A mass $m=25$ is attached to a unit spring with $k=1$, and frictional coefficient $\beta=, 01$. The spring will break when it moves more than 1 unit. Ignoring the effect of the transient, what is the maximum allowable amplitude $\alpha$ of periodic forcing at frequency $\gamma=$
(a) .19 ?
(b) 2 ?
(c) 21 ?

Abhishek Jana
Abhishek Jana
Numerade Educator
03:40

Problem 5

A first order linear system $\dot{u}=a u+b v, \dot{v}=c u+d v$, can be converted into a single second order differential equation by the following device. Assuming that $b \neq 0$, solve the system for $v$ and $\dot{v}$ in terms of $u$ and $\dot{u}$. Then differentiate your equation for $v$ with respect to $t$, and eliminate $\dot{v}$ from the resulting pair of equations. The result is a second order ordinary differential equation for $u(t)$. (a) Write out the second order equation in terms of the coefficients $a, b, c, d$ of the first order system. (b) Show that there is a one-to-one correspondence between solutions of the system and solutions of the scalar differential equation. (c) Use this method to solve the following linear systems, and sketch the resulting phase portraits. (i) $\dot{u}=v, \dot{v}=-u$, (ii) $\dot{u}=2 u+5 v, \dot{v}=-u$, (iii) $\dot{u}=4 u-v$, $\dot{v}=6 u-3 v$, (iv) $\dot{u}=u+v, \dot{v}=u-v$, (v) $\dot{u}=v, \dot{v}=0$. (d) Show how to obtain a second order equation satisfied by $v(t)$ by an analogous device. Are the second order equations for $u$ and for $v$ the same?
(e) Discuss how you might proceed if $b=0$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
09:45

Problem 5

Which of the following $2 \times 2$ systems are gradient flows? Which are Hamiltonian systems? In each case, discuss the stability of the zero solution.
(a)
$$
\dot{u}=-2 u+v \text {, }
$$
$$
\dot{v}=u-2 v \text {, }
$$
(b)
$$
\dot{u}=u-2 v,
$$
(c) $\dot{u}=v$,
(d) $\dot{u}=-v$,
$\dot{v}=-2 u+v$,
$\dot{v}=u$,
(e)
$$
\begin{aligned}
& \dot{u}=-u-2 v, \\
& \dot{v}=-2 u-v .
\end{aligned}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 5

Sketch the phase portrait for the following systems:
(a) $\begin{aligned} & \dot{u}_1=u_1-3 u_2, \\ & \dot{u}_2=-3 u_1+u_2 .\end{aligned}$
(b)
$$
\begin{aligned}
& \dot{u}_1=u_1-4 u_2, \\
& \dot{u}_2=u_1-u_2 .
\end{aligned}
$$
(c) $\dot{u}_1=u_1+u_2$,
(d)
$$
\begin{aligned}
& \dot{u}_1=u_1+u_2, \\
& \dot{u}_2=u_2 .
\end{aligned}
$$
(e) $\dot{u}_1=\frac{3}{2} u_1+\frac{5}{2} u_2$.
$$
\dot{u}_2=-\frac{5}{2} u_1+\frac{3}{2} u_2 \text {. }
$$

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07:35

Problem 5

Find $e^{\mathcal{A}}$ when $A=$
(a) $\left(\begin{array}{rr}5 & -2 \\ -2 & 5\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & -2 \\ 1 & 1\end{array}\right)$,
(c) $\left(\begin{array}{ll}2 & -1 \\ 4 & -2\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -5\end{array}\right)$,
(e) $\left(\begin{array}{rrr}0 & 1 & -2 \\ -1 & 0 & 2 \\ 2 & -2 & 0\end{array}\right)$.

Liliane Martins
Liliane Martins
Numerade Educator
02:26

Problem 5

(a) Determine the natural frequencies of the Newtonian system $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}3 & -2 \\ -2 & 6\end{array}\right) \mathbf{u}=\mathbf{0}$.
(b) What is the dimension of the space of solutions? Explain your answer.
(c) Write out the general solution. (d) For which initial conditions is the resulting motion (i) periodic? (ii) quasi-periodic? (iii) both? (iv) neither? Justify your answer.

James Kiss
James Kiss
Numerade Educator
01:51

Problem 5

(a) For what range of frequencies $\gamma$ can you force the mass in Exercise 10.6.4 with amplitude $\alpha=.5$ without breaking the spring? (b) How large should the friction be so that you can safely force the mass at any frequency?

Jilin Wang
Jilin Wang
Boston University
03:48

Problem 6

(a) Show that if $\mathbf{u}(t)$ solves $\mathbf{u}=A \mathbf{u}$, then $\mathbf{v}(t)=\mathbf{u}(2 t)$ solves $\dot{\mathbf{v}}=B \mathbf{v}$, where $B=2 A$.
(b) How are the solution trajectories of the two systems related?

Ashley Boni
Ashley Boni
Numerade Educator
02:43

Problem 6

(a) Show that the matrix $A=\left(\begin{array}{rrrr}0 & 1 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right)$ has $\lambda= \pm \mathrm{i}$ as incomplete complex conjugate eigenvalues. (b) Find the general real solution to $\mathbf{u}=A \mathbf{u}$.
(c) Explain the behavior of a typical solution. Why is the zero solution not stable?

Sanchit Jain
Sanchit Jain
Numerade Educator
03:40

Problem 6

Which of the 14 possible two-dimensional phase portraits can occur for the phase plane equivalent (10.8) of a second order scalar ordinary differential equation?

Nick Johnson
Nick Johnson
Numerade Educator
00:52

Problem 6

Let $A=\left(\begin{array}{cc}0 & -2 \pi \\ 2 \pi & 0\end{array}\right)$. Show that $e^A=\mathrm{I}$.

Steven Clarke
Steven Clarke
Numerade Educator
00:41

Problem 6

Answer Exercise 10.5.5 for the system $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{ll}73 & 36 \\ 36 & 52\end{array}\right) \mathbf{u}=\mathbf{0}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 6

Suppose the mass-spring-oil system of Exercise $10.5 .39(\mathrm{~b})$ is subject to a periodic external force $2 \cos 2 t$. Discuss, in as much detail as you can, the long-term motion of the mass.

Supratim Pal
Supratim Pal
Numerade Educator
01:02

Problem 7

Let $A$ be a constant $n \times n$ matrix. Let $\mathbf{u}(t)$ be a solution to the system $\frac{d \mathbf{u}}{d t}=A \mathbf{u}$.
(a) Show that its derivatives $\frac{d^k \mathbf{u}}{d t^k}$ for $k=1,2, \ldots$, are also solutions,
(b) Show that $\frac{d^k \mathbf{u}}{d t^k}=A^k \mathbf{u}$.

Nick Johnson
Nick Johnson
Numerade Educator
03:37

Problem 7

Let $A$ be a real $3 \times 3$ matrix, and assume that the linear system $\mathbf{u}=A \mathbf{u}$ has a periodic solution of period $P$. Prove that every periodic solution of the system has period $P$. What other types of solutions can there be? Is the zero solution necessarily stable?

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator

Problem 7

Which of the 14 possible two-dimensional phase portraits can occur
(a) for a linear gradient flow $(10.19)$ ? (b) for a linear Hamiltonian system (10.25)?

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Problem 7

What is $e^{t O}$ where $O$ is the $n \times n$ zero matrix?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 7

Find the general solution to the following second order systems:
(a) $\frac{d^2 u}{d t^2}=-3 u+2 v, \frac{d^2 v}{d t^2}=2 u-3 v$.
(b) $\frac{d^2 u}{d t^2}=-11 u-2 v, \frac{d^2 v}{d t^2}=-2 u-14 v$.
(c) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 9\end{array}\right) \mathbf{u}=\mathbf{0}$
(d) $\frac{d^2 \mathbf{u}}{d t^2}=\left(\begin{array}{rrr}-6 & 4 & -1 \\ 4 & -6 & 1 \\ -1 & 1 & -11\end{array}\right) \mathbf{u}$.

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01:55

Problem 7

Write down the solution $u(t, \gamma)$ to the initial value problem $m \frac{d^2 u}{d t^2}+k u=\alpha \cos \gamma t$, $u(0)=\dot{u}(0)=0$, for (a) a non-resonant forcing function at frequency $\gamma \neq \omega$;
(b) a resonant forcing function at frequency $\gamma=\omega$.
(c) Show that, as $\gamma \rightarrow \omega$, the limit of the non-resonant solution equals the resonant solution. Conclude that the solution $u(t, \gamma)$ depends continuously on the frequency $\gamma$ even though its mathematical formula changes significantly at resonanoe.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:02

Problem 8

True or false: Each solution to a phase plane system moves at a constant speed along its trajectory.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:18

Problem 8

Are the conclusions of Exercise 10.2 .7 valid when $A$ is a $4 \times 4$ matrix?

James Kiss
James Kiss
Numerade Educator
04:54

Problem 8

(a) Solve the initial value problem $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}-1 & 2 \\ -1 & -3\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$.
(b) Sketch a picture of your solution curve $\mathbf{u}(t)$, indicating the direction of motion.
(c) Is the origin (i) stable? (ii) asymptotically stable? (iii) unstable? (iv) none of these? Justify your answer.

Yaw Asomani
Yaw Asomani
Numerade Educator
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Problem 8

Find all matrices $A$ such that $e^{t A}=0$.

Nick Johnson
Nick Johnson
Numerade Educator
01:39

Problem 8

Two masses are connected by three springs to top and bottom supports. Can you find a collection of spring constants $c_1, c_2, c_3$ such that all vibrations are periodic?

Supratim Pal
Supratim Pal
Numerade Educator
00:53

Problem 8

Justify the solution formulas (10.107) and (10.108).

Heather Zimmers
Heather Zimmers
Numerade Educator
03:43

Problem 9

True or false: The phase plane trajectories (10.10) for $\left(c_1, c_2\right)^T \neq 0$ are hyperbolas.

Yujie Wang
Yujie Wang
College of San Mateo

Problem 9

Let $A$ be a real $5 \times 5$ matrix, and assume that $A$ has eigenvalues $\mathrm{i},-\mathrm{i},-2,-1$ (and no others). Is the zero solution to the linear system $\dot{\mathbf{u}}=A \mathbf{u}$ necessarily stable? Explain. Does your answer change if $A$ is $6 \times 6$ ?

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02:30

Problem 9

Explain in detail why the columns of $e^{t A}$ form a basis for the solution space to the system $\dot{\mathbf{u}}=A \mathbf{u}$.

Victor Salazar
Victor Salazar
Numerade Educator
09:31

Problem 9

Suppose the bottom support in the mass-spring chain in Example 10.40 is removed.
(a) Do you predict that the vibration rate will (i) speed up, (ii) slow down, or (iii) stay the same? (b) Verify your prediction by computing the new vibrational frequencies.
(c) Suppose the middle mass is displaced by a unit amount and then let go. Compute and graph the solutions in both situations. Discuss what you observe.

Brian Ketelobeter
Brian Ketelobeter
Numerade Educator
08:05

Problem 9

(a) Does a function of the form $u(t)=a \cos \gamma t-b \cos \omega t$ still exhibit beats when $\gamma \approx \omega$, but $a \neq b$ ? Use a computer to graph some particular cases and discuss what you observe. (b) Explain to what extent the conclusions based on (10.103) do not depend upon the choice of initial conditions (10.102).

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 10

Use a three-dimensional graphics package to plot solution curves $\left(t, u_1(t), u_2(t)\right)^T$ of the phase plane systems in Exercise 10.1.1. Discuss their shape and explain how they are related to the phase plane trajectories.

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01:34

Problem 10

Prove that if $A$ is strictly diagonally dominant and each diagonal entry is negative, then the zero equilibrium solution to the linear system of ordinary differential equations $\dot{\mathbf{u}}=A \mathbf{u}$ is asymptotically stable.

Anand Jangid
Anand Jangid
Numerade Educator
02:26

Problem 10

Let $A$ be a $2 \times 2$ matrix such that $\operatorname{tr} A=0$ and $\delta=\sqrt{\operatorname{det} A}>0$.
(a) Prove that $e^A=(\cos \delta) \mathrm{I}+\frac{\sin \delta}{\delta}$ A. Hint: Use Exercise 8.2.52.
(b) Establish a similar formula when $\operatorname{det} A<0$. (c) What if $\operatorname{det} A=0$ ?

Anthony Ramos
Anthony Ramos
Numerade Educator
05:26

Problem 10

Show that a single mass that is connected to both the top and bottom supports by two springs of stiffnesses $c_1, c_2$ will vibrate in the same manner as if it were connected to only one support by a spring with the combined stiffness $c=c_1+c_2$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:16

Problem 10

Classify the following $R L C$ circuits as (i) underdamped, (ii) critically damped, or (iii) overdamped: (a) $R=1, L=2, C=4$, (b) $R=4, L=3, C=1$,
(c) $R=2, L=3, C=3$, (d) $R=4, L=10, C=2$, (e) $R=1, L=1, C=3$.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:04

Problem 11

Find the solution to the system of differential equations $\frac{d u}{d t}=3 u+4 v, \frac{d v}{d t}=4 u-3 v$, with initial conditions $u(0)=3$ and $v(0)=-2$.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:04

Problem 11

Find the solution to the system of differential equations $\frac{d u}{d t}=3 u+4 v, \frac{d v}{d t}=4 u-3 v$, with initial conditions $u(0)=3$ and $v(0)=-2$.

Wendi Zhao
Wendi Zhao
Numerade Educator

Problem 11

True or false: The system $\dot{\mathbf{u}}=-H_n \mathbf{u}$, where $H_n$ is the $n \times n$ Hilbert matrix (1.72), is asymptotically stable.

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Problem 11

Show that the origin is an asymptotically stable equilibrium solution to $\mathbf{u}=A \mathbf{u}$ if and only if $\lim _{t \rightarrow \infty} e^{t A}=0$.

Victor Salazar
Victor Salazar
Numerade Educator
03:57

Problem 11

(a) Describe, quantitatively and qualitatively, the normal modes of vibration for a mass-spring chain consisting of 3 unit masses, connected to top and bottom supports by unit springs. (b) Answer the same question when the bottom support is removed.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:25

Problem 11

Find the current in each of the unforced $R L C$ circuits in Exercise 10.6 .10 induced by the initial data $u(0)=1, \dot{u}(0)=0$.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:56

Problem 12

Find the general real solution to the following systems of differential equations:
(a)
$$
\text { (a) } \begin{aligned}
\dot{u}_1 & =u_1+9 u_2+ \\
\dot{u}_2 & =u_1+3 u_2 ; \\
\dot{y}_1 & =y_2, \\
\text { (d) } \dot{y}_2 & =3 y_1+2 y_3, \\
\dot{y}_3 & =-y_2 ;
\end{aligned}
$$
(b)
$$
\begin{aligned}
& \dot{x}_1=4 x_1+3 x_2, \quad \text { (c) } \begin{array}{l}
\dot{y}_1=y_1-y_2, \\
\dot{y}_2=2 y_1+3 y_2 ;
\end{array} \\
& \dot{x}_2=3 x_1-4 x_2 ; \quad \dot{u}_1=u_1-3 u_2+11 u_3,
\end{aligned}
$$
(e)
$$
\begin{aligned}
& \dot{x}_2=-x_1+2 x_2+2 x_3, \\
& \dot{x}_3=x_1-4 x_2+2 x_3
\end{aligned}
$$
(f)
$$
\dot{u}_2=2 u_1-6 u_2+16 u_3 \text {. }
$$
$$
\dot{u}_3=u_1-3 u_2+7 u_3 \text {. }
$$

Samriddhi Singh
Samriddhi Singh
Numerade Educator
04:56

Problem 12

Find the general real solution to the following systems of differential equations:
(a)
$$
\begin{aligned}
& \dot{u}_1=u_1+9 u_2 \\
& \dot{u}_2=u_1+3 u_2 \\
& \dot{y}_1=y_2 \\
& \dot{y}_2=3 y_1+2 y_3 \\
& \dot{y}_3=-y_2
\end{aligned}
$$
(d)
$$
\begin{aligned}
& \dot{y}_2=3 y_1+2 y_3 \\
& \dot{y}_3=-y_2
\end{aligned}
$$
(b)
$$
\begin{aligned}
& \dot{x}_1=4 x_1+3 x_2, \quad \text { (c) } \begin{array}{l}
\dot{y}_1=y_1-y_2, \\
\dot{y}_2=2 y_1+3 y_2 ; \\
\dot{x}_2=3 x_1-4 x_2 ;
\end{array} \dot{x}_1=3 x_1-8 x_2+2 x_3, \quad \dot{u}_1=u_1-3 u_2+11 u_3,
\end{aligned}
$$
(c)
$$
\dot{y}_1=y_1-y_2
$$
(e)
$$
\begin{aligned}
& \dot{x}_2=-x_1+2 x_2+2 x_3, \\
& \dot{x}_3=x_1-4 x_2+2 x_3
\end{aligned}
$$
(f)
$$
\begin{aligned}
& \dot{u}_2=2 u_1-6 u_2+16 u_3, \\
& \dot{u}_3=u_1-3 u_2+7 u_3 .
\end{aligned}
$$

Samriddhi Singh
Samriddhi Singh
Numerade Educator
00:53

Problem 12

True or false: If the zero solution of the linear system of differential equations $\dot{\mathbf{u}}=A \mathbf{u}$ is asymptotically stable, so is the zero solution of the linear iterative system $\mathbf{u}^{(k+1)}=A \mathbf{u}^{(k)}$ with the same coefficient matrix.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
View

Problem 12

Let $A$ be a real square matrix and $e^A$ its exponential. Under what conditions does the linear system $\dot{\mathbf{u}}=e^A \mathbf{u}$ have an asymptotically stable equilibrium solution?

Victor Salazar
Victor Salazar
Numerade Educator
01:26

Problem 12

Find the vibrational frequencies for a mass spring chain with $n$ identical masses, connected by $n+1$ identical springs to both top and bottom supports. Is there any sort of limiting behavior as $n \rightarrow \infty$ ? Hint: See Exercise 8.2.48.

Manish Jain
Manish Jain
Numerade Educator
01:39

Problem 12

A circuit with $R=1, L=2, C=4$, includes an alternating current source $F(t)=25 \cos 2 t$. Find the solution to the initial value problem $u(0)=1, \dot{u}(0)=0$.

Amy Jiang
Amy Jiang
Numerade Educator
09:02

Problem 13

Solve the following initial value problems: $(a) \frac{d \mathbf{u}}{d t}=\left(\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right) \mathbf{u}, \mathbf{u}(1)=\left(\begin{array}{l}1 \\ 0\end{array}\right)$;
(b) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{r}-2 \\ 4\end{array}\right)$;
(c) $\frac{d \mathbf{u}}{d \ell}=\left(\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 0\end{array}\right)$;
(d) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}-1 & 3 & -3 \\ 2 & 2 & -7 \\ 0 & 3 & -4\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$;
(e) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}2 & 1 & -6 \\ -1 & 0 & 4 \\ 0 & -1 & -2\end{array}\right) \mathbf{u}, \mathbf{u}(\pi)=\left(\begin{array}{r}2 \\ -1 \\ -1\end{array}\right)$;
(f) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0\end{array}\right) \mathbf{u}, \mathbf{u}(2)=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right) ; \quad(g) \frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrrr}2 & 1 & -1 & 0 \\ -3 & -2 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1\end{array}\right) \mathbf{u}, \quad \mathbf{u}(0)=\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 1\end{array}\right)$.

Ashley Boni
Ashley Boni
Numerade Educator
09:02

Problem 13

Solve the following initial value problems: $(a) \frac{d \mathbf{u}}{d t}=\left(\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right) \mathbf{u}, \mathbf{u}(1)=\left(\begin{array}{l}1 \\ 0\end{array}\right)$;
(b) $\frac{d \mathbf{u}}{d \ell}=\left(\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{r}-2 \\ 4\end{array}\right)$;
(c) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 0\end{array}\right)$;
(d)
$\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}-1 & 3 & -3 \\ 2 & 2 & -7 \\ 0 & 3 & -4\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$;
(e) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}2 & 1 & -6 \\ -1 & 0 & 4 \\ 0 & -1 & -2\end{array}\right) \mathbf{u}, \mathbf{u}(\pi)=\left(\begin{array}{r}2 \\ -1 \\ -1\end{array}\right)$;
(f) $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0\end{array}\right) \mathbf{u}, \mathbf{u}(2)=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right) ;(g) \frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrrr}2 & 1 & -1 & 0 \\ -3 & -2 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 1\end{array}\right)$.

Ashley Boni
Ashley Boni
Numerade Educator
02:34

Problem 13

Let $\mathbf{u}(t)$ solve $\dot{\mathbf{u}}=A \mathbf{u}$. Let $\mathbf{v}(t)=\mathbf{u}(-t)$ be its time reversal.
(a) Write down the linear system $\dot{\mathbf{v}}=B \mathbf{v}$ satisfied by $\mathbf{v}(t)$. Then classify the following statements as true or false. As always, explain your answers. (b) If $\mathbf{u}=A \mathbf{u}$ is asymptotically stable, then $\dot{\mathbf{v}}=B \mathbf{v}$ is unstable. (c) If $\dot{\mathbf{u}}=A \mathbf{u}$ is unstable, then $\dot{\mathbf{v}}=B \mathbf{v}$ is asymptotically stable. (d) If $\mathbf{u}=A \mathbf{u}$ is stable, then $\dot{\mathbf{v}}=B \mathbf{v}$ is stable.

Vipender Yadav
Vipender Yadav
Numerade Educator
00:47

Problem 13

True or false:
(a) $e^{A^{-1}}=\left(e^A\right)^{-1}$
(b) $e^{A+A^{-1}}=e^A e^{A^{-1}}$.

Kian Manafi
Kian Manafi
Numerade Educator
05:43

Problem 13

Suppose the illustrated planar structure has unit masses at the nodes and the bars are all of unit stiffness. (a) Write down the system of differential equations that describes the dynamical vibrations of the structure. (b) How many independent modes of vibration are there? (c) Find numerical values for the vibrational frequencies. (d) Describe what happens when the structure vibrates in each of the normal modes. (e) Suppose the left-hand mass is displaced a unit horizontal distance. Determine the subsequent motion.

Narayan Hari
Narayan Hari
Numerade Educator
05:03

Problem 13

A superconducting $L C$ circuit has no resistance: $R=0$. Discuss what happens when the circuit is wired to an alternating current source $F(t)=\alpha \cos \gamma t$.

NT
Nikhil Tiwari
Numerade Educator
01:47

Problem 14

(a) Find the solution to the system $\frac{d x}{d t}=-x+y, \frac{d y}{d t}=-x-y$, that has initial conditions $x(0)=1, y(0)=0$. (b) Sketch a phase portrait of the system that shows several typical solution trajectories, including the solution you found in part (a). Clearly indicate the direction of increasing $t$ on your curves.

Nick Johnson
Nick Johnson
Numerade Educator
00:09

Problem 14

True or false: (a) If $\operatorname{tr} A>0$, then the system $\mathbf{u}=A \mathbf{u}$ is unstable.
(b) If $\operatorname{det} A>0$, then the system $\dot{\mathbf{u}}=A \mathbf{u}$ is unstable.

Ashley Hanson
Ashley Hanson
Numerade Educator
02:52

Problem 14

Prove formula (10.44). Hint: Fix $s$ and prove that, as functions of $t$, both sides of the equation define matrix solutions with the same initial conditions. Then use uniqueness.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
01:27

Problem 14

When does a homogeneous real first order linear system $\mathbf{u}=A \mathbf{u}$ have a quasi-periodic solution? What is the smallest dimension in which this can occur?

Nick Johnson
Nick Johnson
Numerade Educator
07:18

Problem 14

A circuit with $R=.002, L=12.5$, and $C=50$ can carry a maximum current of 250. Ignoring the effect of the transient, what is the maximum allowable amplitude $\alpha$ of an applied periodic current $F(t)=\alpha \cos \gamma t$ at frequency $\gamma=(a) .04$ ? (b) .05? (c) .1?

Ben Nicholson
Ben Nicholson
Numerade Educator
01:13

Problem 15

A planar steady-state fluid flow has velocity vector field $\mathbf{v}=(2 x-3 y, x-y)^T$ at position $\mathbf{x}=(x, y)^T$. The corresponding fluid motion is described by the differential equation $\frac{d \mathbf{x}}{d t}=\mathbf{v}$. A floating object starts out at the point $(1,1)^T$. Find its position after one time unit.

Anand Jangid
Anand Jangid
Numerade Educator
01:23

Problem 15

True or false: If $K$ is positive semi-definite, then the zero solution to $\mathbf{u}=-K \mathbf{u}$ is stable.

Gregory Higby
Gregory Higby
Numerade Educator
01:01

Problem 15

Prove that $A$ commutes with its exponential: $A e^{t A}=e^{t A} A$.

Nick Johnson
Nick Johnson
Numerade Educator
02:51

Problem 15

Suppose you are given $n$ different springs. In which order should you connect them to unit masses so that the mass-spring chain vibrates the fastest? Does your answer depend upon the relative sizes of the spring constants? Does it depend upon whether the bottom mass is attached to a support or left hanging free? First try the case of three springs with spring stiffnesses $c_1=1, c_2=2, c_3=3$. Then try varying the stiffnesses. Finally, predict what will happen with 4 or 5 springs, and see whether you can make a general conjecture.

Adnan Gill
Adnan Gill
Numerade Educator
03:00

Problem 15

Given the circuit in Exercise 10.6.14, over what range of frequencies $\gamma$ can you supply a unit amplitude periodic current source?

Supratim Pal
Supratim Pal
Numerade Educator
01:44

Problem 16

A steady-state fluid flow has velocity vector field $\mathbf{v}=(-2 y, 2 x, z)^T$ at position $\mathbf{x}=(x, y, z)^T$. Describe the motion of the fluid particles as governed by the differential equation $\frac{d \mathbf{x}}{d t}=\mathbf{v}$.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 16

True or false: If $A$ is a symmetric matrix, then the system $\dot{\mathbf{u}}=-A^2 \mathbf{u}$ has an asymptotically stable equilibrium solution.

Nick Johnson
Nick Johnson
Numerade Educator
00:59

Problem 16

(a) Prove that the exponential of the transpose of a matrix is the transpose of its exponential: $e^{t A^T}=\left(e^{t A}\right)^T$. (b) What does this imply about the solutions to the linear systems $\dot{\mathbf{u}}=A \mathbf{u}$ and $\dot{\mathbf{v}}=A^T \mathbf{v}$ ?

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator

Problem 16

Find the general solution to the following systems. Distinguish between the vibrational and unstable modes. What constraints on the initial conditions ensure that the unstable modes are not excited? (a) $\frac{d^2 u}{d t^2}=-4 u-2 v, \frac{d^2 v}{d t^2}=-2 u-v$.
(b) $\frac{d^2 u}{d t^2}=-u-3 v, \frac{d^2 v}{d t^2}=-3 u-9 v$.
(c) $\frac{d^2 u}{d t^2}=-2 u+v-2 w, \frac{d^2 v}{d t^2}=u-v$,
$\frac{d^2 w}{d t^2}=-2 u-4 w$.
(d) $\frac{d^2 u}{d t^2}=$
$\frac{d^2 v}{d t^2}=u-v+2 w, \frac{d^2 w}{d t^2}=-2 u+2 v-4 w$.

Check back soon!
01:23

Problem 16

How large should the resistanoe in the circuit in Exercise 10.6.14 be so that you can safely apply any unit amplitude periodic current?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:37

Problem 17

Solve the initial value problem $\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rr}-6 & 1 \\ 1 & -6\end{array}\right) \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{l}1 \\ 2\end{array}\right)$. Explain how orthogonality can help.

Ashley Boni
Ashley Boni
Numerade Educator
00:45

Problem 17

Consider the differential equation $\dot{\mathbf{u}}=-K \mathbf{u}$, where $K$ is positive semi-definite.
(a) Find all equilibrium solutions. (b) Prove that all non-constant solutions decay exponentially fast to some equilibrium. What is the decay rate? (c) Is the origin stable, asymptotically stable, or unstable? (d) Prove that, as $t \rightarrow \infty$, the solution $\mathbf{u}(t)$ converges to the orthogonal projection of its initial vector $\mathbf{a}=\mathbf{u}(0)$ onto ker $K$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:14

Problem 17

Prove that if $A=S B S^{-1}$ are similar matrices, then so are $e^{t A}=S e^{t B} S^{-1}$.

Nick Johnson
Nick Johnson
Numerade Educator
02:22

Problem 17

Let $K=\left(\begin{array}{rrr}3 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 3\end{array}\right)$. (a) Find an orthogonal matrix $Q$ and a diagonal matrix $\Lambda$
such that $K=Q \wedge Q^T$. (b) Is $K$ positive definite? (c) Solve the second order system $\frac{d^2 \mathbf{u}}{d t^2}=A \mathbf{u}$ subject to the initial conditions $\mathbf{u}(0)=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right), \frac{d \mathbf{u}}{d t}(0)=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$.
(d) Is your solution periodic? If your answer is yes, indicate the period.
(e) Is the general solution to the system periodic?

Sana Riaz
Sana Riaz
Numerade Educator
01:29

Problem 17

Find the general solution to the following forced second order systems:
(a) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}7 & -2 \\ -2 & 4\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0\end{array}\right)$,
(b) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}5 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}0 \\ 5 \sin 3 t\end{array}\right)$,
(c) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}13 & -6 \\ -6 & 8\end{array}\right) \mathbf{u}=\left(\begin{array}{r}5 \cos 2 t \\ \cos 2 t\end{array}\right)$,
(d) $\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}3 & -1 \\ -1 & 2\end{array}\right) \mathbf{u}=\left(\begin{array}{r}\cos \frac{1}{2} t \\ -\cos \frac{1}{2} t\end{array}\right)$.
(e) $\left(\begin{array}{ll}3 & 0 \\ 0 & 5\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}4 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 11 \sin 2 t\end{array}\right)$,
(f) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0 \\ \cos t\end{array}\right)$

Manik Pulyani
Manik Pulyani
Numerade Educator
14:14

Problem 18

(a) Find the eigenvalues and eigenvectors of $K=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right)$.
(b) Verify that the eigenvectors are mutually orthogonal. (c) Based on part (a), is $K$ positive definite, positive semi-definite, or indefinite? (d) Solve the initial value problem $\frac{d \mathbf{u}}{d t}=K \mathbf{u}, \mathbf{u}(0)=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$, using orthogonality to simplify the computations.

Jason H
Jason H
Numerade Educator
01:49

Problem 18

Suppose that $\mathbf{u}(t)$ satisfies the gradient flow system (10.22).
(a) Prove that $\frac{d}{d t} q(\mathbf{u})=-\|K \mathbf{u}\|^2$.
(b) Explain why if $\mathbf{u}(t)$ is any nonconstant solution to the gradient flow, then $q(\mathbf{u}(t))$ is a strictly decreasing function of $t$, thus quantifying how fast a gradient flow decreases energy.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:17

Problem 18

Prove that $e^{t(A-\lambda I)}=e^{-\ell \lambda} e^{t A}$ by showing that both sides are matrix solutions to the same initial value problem.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:06

Problem 18

Answer Exercise 10.5.17 when $A=\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 1 & -1 \\ 0 & -1 & 2\end{array}\right)$.

Adarsh Kumar
Adarsh Kumar
Numerade Educator
01:43

Problem 18

(a) Find the resonant frequencies of a mass-spring chain consisting of two masses, $m_1=1$ and $m_2=2$, connected to top and bottom supports by identical springs with unit stiffness. (b) Write down an explicit forcing function that will excite the resonance.

James Kiss
James Kiss
Numerade Educator

Problem 19

Demonstrate that one can also solve the initial value problem in Example 10.8 by writing the solution as a complex linear combination of the complex eigensolutions, and then using the initial conditions to specify the coefficients.

Check back soon!

Problem 19

Let $H(u, v)=a u^2+b u v+c v^2$ be a quadratic function. (a) Prove that the nonequilibrium trajectories of the associated Hamiltonian system and those of the gradient flow are mutually orthogonal, i.e., they always intersect at right angles. (b) Verify this result for the particular quadratic functions (i) $u^2+3 v^2$, (ii) $u v$, by drawing representative trajectories of both systems on the same graph.

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02:26

Problem 19

Let $A$ be a real matrix. (a) Explain why $e^A$ is a real matrix. (b) Prove that det $e^A>0$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:47

Problem 19

Compare the solutions to the mass-spring system (10.65) with tiny spring constant $k=\varepsilon \& 1$ to those of the completely unrestrained system (10.78). Are they close? Discuss.

James Kiss
James Kiss
Numerade Educator
01:24

Problem 19

Suppose one of the fixed supports is removed from the mass-spring chain of Exercise 10.6.18. Does your forcing function still excite the resonance? Do the internal vibrations of the masses ( $i$ ) speed up, (ii) slow down, or (iii) remain the same? Does your answer depend upon which of the two supports is removed?

Supratim Pal
Supratim Pal
Numerade Educator
03:14

Problem 20

Determine whether the following vector-valued functions are linearly dependent or linearly independent:
(a) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{r}-t \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{c}1+t \\ t\end{array}\right),\left(\begin{array}{c}1-t^2 \\ t-t^2\end{array}\right)$
(c) $\left(\begin{array}{l}1 \\ t\end{array}\right),\left(\begin{array}{l}t \\ 2\end{array}\right),\left(\begin{array}{r}-t \\ t\end{array}\right)$,
(d)
$\left(\begin{array}{c}e^{-t} \\ -e^t\end{array}\right),\left(\begin{array}{c}-e^{-t} \\ e^t\end{array}\right)$
(e) $\left(\begin{array}{r}e^{2 t} \cos 3 t \\ -e^{2 t} \sin 3 t\end{array}\right),\left(\begin{array}{c}e^{2 t} \sin 3 t \\ e^{2 t} \cos 3 t\end{array}\right)$,
(f) $\left(\begin{array}{c}\cos 3 t \\ \sin 3 t\end{array}\right),\left(\begin{array}{c}\sin 3 t \\ \cos 3 t\end{array}\right)$,
(g) $\left(\begin{array}{c}1 \\ t \\ 1-t\end{array}\right),\left(\begin{array}{c}0 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{c}3 \\ 1+3 t \\ 2-3 t\end{array}\right)$.
(h) $\left(\begin{array}{r}e^t \\ -e^t \\ e^t\end{array}\right),\left(\begin{array}{r}e^t \\ e^t \\ -e^t\end{array}\right)+\left(\begin{array}{r}-e^t \\ e^t \\ e^t\end{array}\right)$,
(i) $\left(\begin{array}{c}e^t \\ t e^t \\ t^2 e^t\end{array}\right),\left(\begin{array}{c}t^2 e^t \\ e^t \\ t e^t\end{array}\right),\left(\begin{array}{c}t e^t \\ t^2 e^t \\ e^t\end{array}\right),\left(\begin{array}{c}e^t \\ e^t \\ e^t\end{array}\right)$.

Christian Otero
Christian Otero
Numerade Educator
00:39

Problem 20

True or false: If the Hamiltonian system for $H(u, v)$ is stable, then the corresponding gradient flow $\dot{\mathbf{u}}=-\nabla H$ is stable.

Lucas Finney
Lucas Finney
Numerade Educator
04:47

Problem 20

Show that $\operatorname{tr} A=0$ if and only if $\operatorname{det} e^{t A}=1$ for all $t$.

Scott Stetson
Scott Stetson
Numerade Educator
02:27

Problem 20

Discuss the three-dimensional motions of the triatomic molecule of Example 10.41.
Are the vibrational frequencies the same as those of the one-dimensional model?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:00

Problem 20

Find the resonant frequencies of the following structures, assuming the nodes all have unit mass. Then find a means of forcing the structure at one of the resonant frequencies, and yet not exciting the resonance. Can you also force the structure without exciting any mechanism or rigid motion? (a) The square truss of Exercise 6.3 .5 ; (b) the joined square truss of Exercise 6.3.6; (c) the house of Exercise 6.3.8; (d) the triangular space station of Example 6.6; (e) the triatomic molecule of Example 10.41; $(f)$ the water molecule of Exercise 10.5.30.

James Kiss
James Kiss
Numerade Educator
04:23

Problem 21

Let $A$ be a constant matrix. Suppose $\mathbf{u}(t)$ solves the initial value problem $\mathbf{u}=A \mathbf{u}$, $\mathbf{u}(0)=\mathbf{b}$. Prove that the solution to the initial value problem $\dot{\mathbf{u}}=A \mathbf{u}, \mathbf{u}\left(t_0\right)=\mathbf{b}$, is equal to $\overline{\mathbf{u}}(t)=\mathbf{u}\left(t-t_0\right)$. How are the solution trajectories related?

Christian Otero
Christian Otero
Numerade Educator
00:37

Problem 21

True or false: A nonzero linear $2 \times 2$ gradient flow cannot be a Hamiltonian flow:

Monica Miller
Monica Miller
Numerade Educator
View

Problem 21

Justify the matrix Leibniz rule (10.41) using the formula for matrix multiplication.

Victor Salazar
Victor Salazar
Numerade Educator
02:06

Problem 21

So far, our mass-spring chains have been allowed to move only in the vertical direction. (a) Set up the system governing the planar motions of a mass-spring chain consisting of two unit masses attached to top and bottom supports by unit springs, where the masses are allowed to move in the longitudinal and transverse directions. Compare the resulting vibrational frequencies with the one-dimensional case. (b) Repeat the analysis when the bottom support is removed. (c) Can you make any conjectures concerning the planar motions of general mass-spring chains?

Penny Riley
Penny Riley
Numerade Educator
01:26

Problem 22

Suppose $\mathbf{u}(t)$ and $\overline{\mathbf{u}}(t)$ both solve the linear system $\dot{\mathbf{u}}=A \mathbf{u}$. (a) Suppose they have the same value $\mathbf{u}\left(t_1\right)=\overrightarrow{\mathbf{u}}\left(t_1\right)$ at any one time $t_1$. Show that they are, in fact, the same solution: $\mathbf{u}(t)=\overline{\mathbf{u}}(t)$ for all $t$. (b) What happens if $\mathbf{u}\left(t_1\right)=\overline{\mathbf{u}}\left(t_2\right)$ for some $t_1 \neq t_2$ ? Hint: See Exercise 10.1.21.

Hoan Nguyen
Hoan Nguyen
Numerade Educator

Problem 22

The law of conservation of energy states that the energy in a Hamiltonian system is constant on solutions. (a) Prove that if $\mathbf{u}(t)$ satisfies the Hamiltonian system (10.23), then $H(\mathbf{u}(t))=c$ is a constant, and hence solutions $\mathbf{u}(t)$ move along the level sets of the Hamiltonian or energy function. Explain how the value of $c$ is determined by the initial conditions. (b) Plot the level curves of the particular Hamiltonian function $H(u, v)=u^2-2 u v+2 v^2$ and verify that they coincide with the solution trajectories.

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00:59

Problem 22

Prove that if $U(t)$ is any matrix solution to $\frac{d U}{d t}=A U$, then so is $\bar{U}(t)=U(t) C$, where $C$ is any constant matrix (of compatible size).

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
View

Problem 22

Find the vibrational frequencies and instabilities of the following structures, assuming they have unit masses at all the nodes. Explain in detail how each normal mode moves the structure: (a) the three bar planar structure in Figure 6.13 ; (b) its reinforced version in Figure 6.16; (c) the swing set in Figure 6.18.

Victor Salazar
Victor Salazar
Numerade Educator
04:17

Problem 23

Prove that the general solution to a linear system $\mathbf{u}=\Lambda \mathbf{u}$ with diagonal coefficient matrix $\Lambda=\operatorname{diag}\left(\lambda_1, \ldots+\lambda_n\right)$ is given by $\mathbf{u}(t)=\left(c_1 e^{\lambda_1 t}, \ldots, c_n e^{\lambda_n t}\right)^T$.

Andrija Isakov
Andrija Isakov
Numerade Educator
00:37

Problem 23

True or false: A nonzero linear $2 \times 2$ gradient flow cannot be a Hamiltonian system.

Monica Miller
Monica Miller
Numerade Educator
08:58

Problem 23

Prove that if $A=\left(\begin{array}{cc}B & 0 \\ O & C\end{array}\right)$ is a block diagonal matrix, then so is $e^{t A}=\left(\begin{array}{cc}e^{t B} & 0 \\ O & e^{t C}\end{array}\right)$.

Chris Trentman
Chris Trentman
Numerade Educator
07:00

Problem 23

Assuming unit masses at the nodes, find the vibrational frequencies and describe the normal modes for the following planar structures. What initial conditions will not excite its instabilities? (a) An equilateral triangle; (b) a square; (c) a regular hexagon.

Tianyu Li
Tianyu Li
Numerade Educator
06:35

Problem 24

Show that if $\mathbf{u}(t)$ is a solution to $\mathbf{u}=A \mathbf{u}$, and $S$ is a constant, nonsingular matrix of the same size as $A$, then $\mathbf{v}(t)=S \mathbf{u}(t)$ solves the linear system $\mathbf{v}=B \mathbf{v}$, where $B=S A S^{-1}$ is similar to $A$.

Uma Kumari
Uma Kumari
Numerade Educator
04:11

Problem 24

(a) Explain how to solve the inhomogeneous system $\frac{d \mathbf{u}}{d t}=A \mathbf{u}+\mathbf{b}$ when $\mathbf{b}$ is a constant vector belonging to img $A$. Hint: Look at $\mathbf{v}(t)=\mathbf{u}(t)-\mathbf{u}^*$ where $\mathbf{u}^*$ is an equilibrium solution. (b) Use your method to solve
(i) $\frac{d u}{d t}=u-3 v+1, \frac{d v}{d t}=-u-v$,
(ii) $\frac{d u}{d t}=4 v+2, \frac{d v}{d t}=-u-3$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator

Problem 24

(a) Prove that if $J_{0, n}$ is an $n \times n$ Jordan block matrix with 0 diagonal entries,
$$
\text { cf. (8.49), then } e^{t J_{0, n}}=\left(\begin{array}{cccccc}
1 & t & \frac{t^2}{2} & \frac{t^3}{6} & \cdots & \frac{t^n}{n !} \\
0 & 1 & t & \frac{t^2}{2} & \cdots & \frac{t^{n-1}}{(n-1) !} \\
0 & 0 & 1 & t & \cdots & \frac{t^{n-2}}{(n-2) !} \\
\vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & 1 & t \\
0 & 0 & 0 & \cdots & 0 & 1
\end{array}\right) \text {. }
$$
(b) Determine the exponential of a general Jordan block matrix $J_{\lambda, n^*}$ Hint: Use Exercise 10.4.18. (c) Explain how you can use the Jordan canonical form to compute the exponential of a matrix. Hint: Use Exercise 10.4.23.

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01:59

Problem 24

Answer Exercise 10.5.23 for the three-dimensional motions of a regular tetrahedron.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
View

Problem 25

(i) Combine Exercises 10.1.23 24 to show that if $A=S \Lambda S^{-1}$ is diagonalizable, then the solution to $\dot{\mathbf{u}}=A \mathbf{u}$ can be written as $\mathbf{u}(t)=S\left(c_1 e^{\lambda_1 t}, \ldots, c_n e^{\lambda_n t}\right)^T$, where $\lambda_1, \ldots, \lambda_n$ are its eigenvalues and $S=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the corresponding matrix of eigenvectors.
(ii) Write the general solution to the systems in Exercise 10.1.13 in this form.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 25

Prove Lemma 10.15.

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01:18

Problem 25

Diagonalization provides an alternative method for computing the exponential of a complete matrix. (a) First show that if $D=\operatorname{diag}\left(d_1, \ldots, d_n\right)$ is a diagonal matrix, so is $e^{t D}=\operatorname{diag}\left(e^{t d_1}, \ldots, e^{t d_n}\right)$. (b) Second, using Exercise 10.4.17, prove that if $A=S D S^{-1}$ is diagonalizable, so is $e^{t A}=S e^{t D} S^{-1}$. (c) When possible, use diagonalization to compute the exponentials of the matrices in Exercises 10.4.1-2.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:01

Problem 25

(a) Show that if a structure contains all unit masses and bars with unit stiffness, $c_i=1$, then its frequencies of vibration are the nonzero singular values of the reduced incidence matrix. (b) How would you recognize when a structure is close to being unstable?

James Kiss
James Kiss
Numerade Educator
03:27

Problem 26

Find the general solution to the linear system $\frac{d \mathbf{u}}{d t}=A \mathbf{u}$ for the following incomplete coefficient matrices:
(a) $\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{ll}2 & -1 \\ 9 & -4\end{array}\right)$,
(c) $\left(\begin{array}{rr}-1 & -1 \\ 4 & -5\end{array}\right)$.
(d) $\left(\begin{array}{rrr}4 & -1 & -3 \\ -2 & 1 & 2 \\ 5 & -1 & -4\end{array}\right)$,
(e) $\left(\begin{array}{rrr}-3 & 1 & 0 \\ 1 & -3 & -1 \\ 0 & 1 & -3\end{array}\right)$
(f) $\left(\begin{array}{rrrr}3 & 1 & 1 & 1 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & -1\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}0 & 1 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right)$.

Sherrie Fenner
Sherrie Fenner
Numerade Educator

Problem 26

Prove Proposition 10.18.

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08:26

Problem 26

(a) Prove that if $\lambda$ is an eigenvalue of $A$, then $e^{t \lambda}$ is an eigenvalue of $e^{t A}$. What is the eigenvector? (b) Show that the eigenvalues have the same multiplicities.
Hint: Combine the Jordan canonical form (8.51) with Exercises 10.4.24 and 10.4.25.

Tamara Worner
Tamara Worner
Numerade Educator
04:10

Problem 26

Prove that if the initial velocity satisfies $\dot{\mathbf{u}}\left(t_0\right)=\mathbf{b} \in$ coimg $A$, then the solution to the initial value problem $(10.70,76)$ remains bounded.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
02:53

Problem 27

Find a first order system of ordinary differential equations that has the indicated vector-valued function as a solution:
(a) $\left(\begin{array}{c}e^t+e^{2 t} \\ 2 e^t\end{array}\right)$,
(b) $\left(\begin{array}{c}e^{-t} \cos 3 t \\ -3 e^{-t} \sin 3 t\end{array}\right)$,
(c) $\left(\begin{array}{c}1 \\ t-1\end{array}\right)$,
(d) $\left(\begin{array}{c}\sin 2 t-\cos 2 t \\ \sin 2 t+3 \cos 2 t\end{array}\right)$,
(e) $\left(\begin{array}{c}e^{2 t} \\ e^{-3 t} \\ e^{2 t}-e^{-3 t}\end{array}\right)$,
(f) $\left(\begin{array}{c}\sin t \\ \cos t \\ 1\end{array}\right)$,
(g) $\left(\begin{array}{c}t \\ 1-t^2 \\ 1+t\end{array}\right)$,
(h) $\left(\begin{array}{c}e^t \sin t \\ 2 e^t \cos t \\ e^t \sin t\end{array}\right)$.

Christian Otero
Christian Otero
Numerade Educator

Problem 27

Let $A$ be a symmetric matrix with Spectral Decomposition
$$
A=\lambda_1 P_1+\lambda_2 P_2+\cdots+\lambda_k P_k \text {, }
$$
as in (8.37). Prove that
$$
e^{t A}=e^{t \lambda_1} P_1+e^{\ell \lambda_2} P_2+\cdots+e^{t \lambda_k} P_k
$$

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03:38

Problem 27

Find the general solution to the system (10.82) for the following matrix pairs:
(a)
$M=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right), K=\left(\begin{array}{rr}3 & -1 \\ -1 & 2\end{array}\right)$
(b)
(c)
$M=\left(\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right), K=\left(\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right)$,
(d)
$M=\left(\begin{array}{ll}3 & 0 \\ 0 & 5\end{array}\right), \quad K=\left(\begin{array}{rr}4 & -2 \\ -2 & 3\end{array}\right)$,
(e)
$M=\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right), K=\left(\begin{array}{rr}3 & -1 \\ -1 & 3\end{array}\right)$.
$M=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 6\end{array}\right), K=\left(\begin{array}{rrr}5 & -1 & -1 \\ -1 & 6 & 3 \\ -1 & 3 & 9\end{array}\right)$
(f)
$M=\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 1\end{array}\right), \quad K=\left(\begin{array}{lll}1 & 2 & 0 \\ 2 & 8 & 2 \\ 0 & 2 & 1\end{array}\right)$.

Jingyun Wang
Jingyun Wang
Numerade Educator
01:19

Problem 28

Which sets of functions in Exercise 10.1 .20 can be solutions to a common first order, homogeneous, constant coefficient linear system of ordinary differential equations? If so, find a system they satisfy; if not, explain why not.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
04:42

Problem 28

(a) Show that $U(t)$ satisfies the matrix differential equation $\dot{U}=U B$ if and only if $U(t)=C e^{t B}$, where $C=U(0)$. (b) If $U(0)$ is nonsingular, then $U(t)$ also satisfies a matrix differential equation of the form $\dot{U}=A U$. Is $A=B$ ? Hint: Use Exercise 10,4.17.

Elham Kordzadeh
Elham Kordzadeh
Numerade Educator
08:24

Problem 28

A mass-spring chain consists of two masses, $m_1=1$ and $m_2=2$, connected to top and bottom supports by identical springs with unit stiffness. The upper mass is displaced by a unit distance. Find the subsequent motion of the system.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
03:59

Problem 29

Solve the third order equation $\frac{d^3 u}{d t^3}+3 \frac{d^2 u}{d t^2}+4 \frac{d u}{d t}+12 u=0$ by converting it into a first order system. Compare your answer with what you found in Exercise 10.1.2.

Foster Wisusik
Foster Wisusik
Numerade Educator
01:07

Problem 29

True or false: The solution to the non-autonomous initial value problem $\dot{\mathbf{u}}=A(t) \mathbf{u}, \mathbf{u}(0)=\mathbf{b}, \quad$ is $\quad \mathbf{u}(t)=\exp \left(\int_0^t A(s) d s\right) \mathbf{b}$.

Suzanne W.
Suzanne W.
Numerade Educator
01:29

Problem 29

Answer Exercise 10.5.28 when the bottom support is removed.

Narayan Hari
Narayan Hari
Numerade Educator
01:50

Problem 30

Solve the second order coupled system of ordinary differential equations $\ddot{u}=\dot{u}+u-v$, $\ddot{v}=\dot{v}-u+v$, by converting it into a first order system involving four variables.

Christian Otero
Christian Otero
Numerade Educator
00:54

Problem 30

(a) Suppose $\mathbf{u}_1(t), \ldots, \mathbf{u}_n(t)$ are vector-valued functions whose values at each point $t$ are linearly independent vectors in $\mathbb{R}^n$. Show that they form a basis for the solution space of a homogeneous constant coefficient linear system $\mathbf{u}=A \mathbf{u}$ if and only if each $d \mathbf{u}_j / d t$ is a linear combination of $\mathbf{u}_1(t), \ldots, \mathbf{u}_n(t)$. Hint: Use Exercise 10.4.28. (b) Show that a function $\mathbf{u}(t)$ belongs to the solution space of a homogeneous constant coefficient linear system $\dot{\mathbf{u}}=A \mathbf{u}$ if and only if $\frac{d^n \mathbf{u}}{d t^n}$ is a linear combination of $\mathbf{u}, \frac{d \mathbf{u}}{d t}, \ldots, \frac{d^{n-1} \mathbf{u}}{d t^{n-1}}$. Hint: Use Exercise 10.1.7.

Victor Salazar
Victor Salazar
Numerade Educator
01:22

Problem 30

(a) A water molecule consists of two hydrogen atoms connected at an angle of $105^{\circ}$ to an oxygen atom whose relative mass is 16 times that of each of the hydrogen atoms. If the molecular bonds are modeled as linear unit springs, determine the fundamental frequencies and describe the corresponding vibrational modes. (b) Do the same for a carbon tetrachloride molecule, in which the chlorine atoms, with atomic weight 35 , are positioned on the vertices of a regular tetrahedron and the carbon atom, with atomic weight 12 , is at the center. (c) Finally try a benzene molecule, consisting of 6 carbon atoms arranged in a regular hexagon. In this case, every other bond is double strength because two electrons are shared. (Ignore the six extra hydrogen atoms for simplicity.)

Lottie Adams
Lottie Adams
Numerade Educator
02:59

Problem 31

Suppose that $\mathbf{u}(t) \in \mathbb{R}^n$ is a polynomial solution to the constant coefficient linear system $\mathbf{u}=A \mathbf{u}$. What is the maximal possible degree of $\mathbf{u}(t)$ ? What can you say about $A$ when $\mathbf{u}(t)$ has maximal degree?

Allison Knapp
Allison Knapp
Numerade Educator
17:18

Problem 31

By a (natural) logarithm of a matrix $B$ we mean a matrix $A$ such that $e^A=B$.
(a) Explain why only nonsingular matrices can have a logarithm.
(b) Comparing Exercises 10.4.6-7, explain why the matrix logarithm is not unique.
(c) Find all real logarithms of the $2 \times 2$ identity matrix $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$. Hint: Use Exercise 10.4.26.

Chris Trentman
Chris Trentman
Numerade Educator
01:27

Problem 31

Repeat Exercise 10.5.21 for fully 3-dimensional motions of the chain.

Adnan Gill
Adnan Gill
Numerade Educator
03:26

Problem 32

(a) Under the assumption that $\mathbf{u}_1, \ldots, \mathbf{u}_k$ form a Jordan chain for the coefficient matrix $A$, prove that the functions $(10.17)$ are solutions to the system $\dot{\mathbf{u}}=A \mathbf{u}$.
(b) Prove that they are linearly independent.

Linh Vu
Linh Vu
Numerade Educator
04:12

Problem 32

Find the one-parameter groups generated by the following matrices and interpret geometrically: What are the trajectories? What are the fixed points?
(a) $\left(\begin{array}{ll}2 & 0 \\ 0 & 0\end{array}\right)$.
(b) $\left(\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rr}0 & 3 \\ -3 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rr}0 & -1 \\ 4 & 0\end{array}\right)$,
(e) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$

Supratim Pal
Supratim Pal
Numerade Educator
03:30

Problem 32

Suppose you have masses $m_1=1, m_2=2, m_3=3$, connected to top and bottom supports by identical unit springs. Does rearranging the order of the masses change the fundamental frequencies? If so, which order produces the fastest vibrations?

Shoukat Ali
Shoukat Ali
Other Schools
04:12

Problem 33

Write down the one-parameter groups generated by the following matrices and interpret. What are the trajectories? What are the fixed points?
(a) $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right)$,
(b) $\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & 0 & -2 \\ 0 & 0 & 0 \\ 2 & 0 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{array}\right)$.

Supratim Pal
Supratim Pal
Numerade Educator
03:22

Problem 33

Suppose $M$ is a nonsingular matrix. Prove that $\lambda$ is a generalized eigenvalue of the matrix pair $K, M$ if and only if it is an ordinary eigenvalue of the matrix $P=M^{-1} K$. How are the eigenvectors related? How are the characteristic equations related?

Andrija Isakov
Andrija Isakov
Numerade Educator
View

Problem 34

(a) Find the one-parameter group of rotations generated by the skew-symmetric matrix $A=\left(\begin{array}{rrr}0 & 1 & 1 \\ -1 & 0 & -1 \\ -1 & 1 & 0\end{array}\right)$.
(b) As noted above, $e^{t A}$ represents a family of rotations around a fixed axis in $\mathbb{R}^3$. What is the axis?

Victor Salazar
Victor Salazar
Numerade Educator
10:58

Problem 34

Suppose that $\mathbf{u}(t)$ is a solution to (10.82). Let $N=\sqrt{M}$ denote the positive definite square root of the mass matrix $M$, as defined in Exercise 8.5.27. (a) Prove that the "weighted" displacement vector $\mathbf{u}(t)=N \mathbf{u}(t)$ solves $d^2 \mathbf{u} / d t^2=-\widetilde{K} \mathbf{u}$, where $\bar{K}=N^{-1} K N^{-1}$ is a symmetric, positive semi-definite matrix. (b) Explain in what sense this can serve as an alternative to the generalized eigenvector solution method.

Chris Trentman
Chris Trentman
Numerade Educator
05:56

Problem 35

Choose two of the groups in Exercise 10.4.32 or 10.4-33, and determine whether or not they commute by looking at their infinitesimal generators. Then verify your conclusion by directly computing the commutator of the corresponding matrix exponentials.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 35

Provide the details of the proof of Theorem 10.42.

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03:42

Problem 36

(a) Prove that the commutator of two upper triangular matrices is upper triangular.
(b) Prove that the commutator of two skew-symmetric matrices is skew symmetric.
(c) Is the commutator of two symmetric-matrices symmetric?

Urvashi Arora
Urvashi Arora
Numerade Educator
01:19

Problem 36

State and prove the counterpart of Theorem 10.42 for the variable mass system (10.82).

Nick Johnson
Nick Johnson
Numerade Educator
03:54

Problem 37

Prove that the Jacobi identity
$$
[[A, B], C]+[[C, A], B]+[[B, C], A]=\mathrm{O}
$$
is valid for three $n \times n$ matrices $A, B, C$.

Lucas Finney
Lucas Finney
Numerade Educator
10:24

Problem 37

Consider the overdamped mass -spring equation $\ddot{u}+6 \dot{u}+5 u=0$. If the mass starts out a distance 1 away from equilibrium, how large must the initial velocity be in order that it pass through equilibrium once?
Cuabuoas Thasciag earn

Mike Gaerlan
Mike Gaerlan
Numerade Educator
00:56

Problem 38

Let $0 \neq \mathrm{v} \in \mathbb{R}^3$. (a) Show that the cross product $L_{\mathbf{v}}[\mathbf{x}]=\mathbf{v} \times \mathbf{x}$ defines a linear transformation on $\mathbb{R}^3$. (b) Find the $3 \times 3$ matrix representative $A_{\mathrm{v}}$ of $L_{\mathrm{v}}$ and show that it is skew-symmetric. (c) Show that every non-zero skew-symmetric $3 \times 3$ matrix defines such a cross product map. (d) Show that $\operatorname{ker} A_{\mathbf{y}}$ is spanned by $\mathbf{v}$. (e) Justify the fact that the matrix exponentials $e^{t A_v}$ are rotations around the axis $\mathbf{v}$. Thus, the cross product with a vector serves as the infinitesimal generator of the one-parameter group of rotations around $\mathbf{v}$.

Arun Bana
Arun Bana
Numerade Educator
03:59

Problem 38

Solve the following mass-spring initial value problems, and classify as to (i) overdamped, (ii) critically damped, (iii) underdamped, or (iv) undamped:
(a) $\ddot{u}+6 \dot{u}+9 u=0, u(0)=0, \dot{u}(0)=1$. (b) $\ddot{u}+2 \dot{u}+10 u=0, u(0)=1, \dot{u}(0)=1$.
(c) $\ddot{u}+16 u=0, u(1)=0, \dot{u}(1)=1$. (d) $\ddot{u}+3 \dot{u}+9 u=0, u(0)=0, \dot{u}(0)=1$.
(e) $2 \ddot{u}+3 \dot{u}+u=0, u(0)=2, \dot{u}(0)=0$. (f) $\ddot{u}+6 \dot{u}+10 u=0, u(0)=3, \dot{u}(0)=-2$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
02:30

Problem 39

Given a unit vector $\|\mathbf{u}\|=1$ in $\mathrm{R}^3$, let $A=A_{\mathbf{u}}$ be the corresponding skew-symmetric $3 \times 3$ matrix that satisfies $A \mathbf{x}=\mathbf{u} \times \mathbf{x}$, as in Exercise 10.4.38. (a) Prove the Euler-Rodrigues formula $e^{t A}=\mathrm{I}+(\sin t) A+(1-\cos t) A^2$. Hint: Use the matrix exponential series $(10.47)$.
(b) Show that $e^{t A}=I$ if and only if $t$ is an integer multiple of $2 \pi$. (c) Generalize parts (a) and (b) to a non-unit vector $\mathbf{v} \neq \mathbf{0}$.

Victor Salazar
Victor Salazar
Numerade Educator
00:44

Problem 39

(a) A mass weighing 16 pounds stretches a spring 6.4 feet. Assuming no friction, determine the equation of motion and the natural frequency of vibration of the mass-spring system. Use the value $g=32 \mathrm{ft} / \mathrm{sec}^2$ for the gravitational acceleration. (b) The massspring system is placed in a jar of oil, whose frictional resistance equals the speed of the mass. Assume the spring is stretched an additional 2 feet from its equilibrium position and let go. Determine the motion of the mass. (c) Is the system overdamped or underdamped? Are the vibrations more rapid or less rapid than in the undamped system?

Adnan Gill
Adnan Gill
Numerade Educator
04:47

Problem 40

Let $A=\left(\begin{array}{rrr}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), \mathbf{b}=\left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)$. (a) Show that the solution to the linear system $\dot{\mathbf{x}}=A \mathbf{x}$ represents a rotation of $\mathbb{R}^3$ around the $z$-axis. What is the trajectory of a point $\mathbf{x}_0$ ? (b) Show that the solution to the inhomogeneous system $\dot{\mathbf{x}}=A \mathbf{x}+\mathbf{b}$ represents a screw motion of $\mathbf{R}^3$ around the z-axis. What is the trajectory of a point $\mathbf{x}_0$ ? (c) More generally, given $0 \neq \mathbf{a} \in \mathbb{R}^3$, show that the solution to $\dot{\mathbf{x}}=\mathbf{a} \times \mathbf{x}+\mathbf{a}$ represents a family of screw motions along the axis a.

Arwa Ali
Arwa Ali
Numerade Educator
06:35

Problem 40

Suppose you convert the second order equation (10.87) into its phase plane equivalent. What are the phase portraits corresponding to (a) undamped, (b) underdamped, (c) critically damped, and (d) overdamped motion?

Mike Gaerlan
Mike Gaerlan
Numerade Educator
02:38

Problem 41

Let $A$ be an $n \times n$ matrix whose last row has all zero entries. Prove that the last row of $e^{t A}$ is $\mathbf{e}_n^T=(0, \ldots, 0,1)$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:30

Problem 41

(a) Prove that, given a non-constant solution to an overdamped mass-spring system, there is at most one time where $u\left(t_{\star}\right)=0$. (b) Is this statement also valid in the critically damped case?

Mike Gaerlan
Mike Gaerlan
Numerade Educator
17:18

Problem 42

Let $A=\left(\begin{array}{ll}B & \mathbf{c} \\ 0 & 0\end{array}\right)$ be in block form, where $B$ is an $n \times n$ matrix, $\mathbf{c} \in \mathbb{R}^n$, while $\mathbf{0}$ denotes the zero row vector with $n$ entries. Show that its matrix exponential is also in block form $e^{t A}=\left(\begin{array}{cc}e^{t B} & \mathbf{f}(t) \\ \mathbf{0} & 1\end{array}\right)$. Can you find a formula for $\mathbf{f}(t)$ ?

Chris Trentman
Chris Trentman
Numerade Educator
01:08

Problem 42

Discuss the possible behaviors of a mass moving in a frictional medium that is not attached to a spring, i.e., set $k=0$ in (10.87).

Lucas Finney
Lucas Finney
Numerade Educator
02:10

Problem 43

According to Exercise 7.3.10, an $(n+1) \times(n+1)$ matrix of the block form $\left(\begin{array}{cc}A & \mathbf{b} \\ \mathbf{0} & 1\end{array}\right)$ in which $A$ is an $n \times n$ matrix and $\mathbf{b} \in \mathbb{R}^n$ can be identified with the affine transformation $F[\mathbf{x}]=A \mathbf{x}+\mathbf{b}$ on $\mathbb{R}^n$. Exercise 10.4 .42 shows that every matrix in the one-parameter group $e^{t B}$ generated by $B=\left(\begin{array}{ll}A & b \\ 0 & 0\end{array}\right)$ has such a form, and hence we can identify $e^{t B}$ as a family of affine maps on $\mathbb{R}^n$. Describe the affine transformations of $\mathbb{R}^2$ generated by the following matrices:
(a) $\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$.
(b) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & -1 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 0 & 1 \\ 0 & -1 & -2 \\ 0 & 0 & 0\end{array}\right)$

Ashley Boni
Ashley Boni
Numerade Educator
05:43

Problem 44

(a) Given a homogeneous linear dynamical system with invariant stable, unstable, and center subspaces $S, U, C$, explain why the origin is asymptotically stable if and only if $C=U=\{0\}$. (b) Is the origin stable if $U=\{0\}$ but $C \neq\{0\}$ ?

Uma Kumari
Uma Kumari
Numerade Educator
04:10

Problem 45

Find the (real) stable, unstable, and center subspaces of the following linear systems:
(a)
$$
\begin{aligned}
& \dot{u}_1=9 u_2 \text {, } \\
& \dot{u}_2=-u_1 \text {; } \\
& \dot{x}_2=3 x_1 \text {; } \\
& \dot{u}_1=u_1-3 u_2+11 u_3, \\
&
\end{aligned}
$$
(b) $\dot{x}_1=4 x_1+x_2$,
(e)
$$
\begin{aligned}
& \dot{u}_2=2 u_1-6 u_2+16 u_3, \\
& \dot{u}_3=u_1-3 u_2+7 u_3,
\end{aligned} \quad \text { (f) } \frac{d \mathbf{u}}{d \ell}=\left(\begin{array}{rrr}
-1 & 3 & -3 \\
2 & 2 & -7 \\
0 & 3 & -4
\end{array}\right) \mathbf{u}
$$
(f)
$$
\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}
-1 & 3 & -3 \\
2 & 2 & -7 \\
0 & 3 & -4
\end{array}\right) \mathbf{u},
$$
(c)
$$
\begin{aligned}
& \dot{y}_1=y_1-y_2 \\
& \dot{y}_2=2 y_1+3 y_2
\end{aligned}
$$
$$
\dot{z}_1=z_2 \text {, }
$$
(d)
$$
\begin{aligned}
& \dot{z}_2=3 z_1+2 z_3, \\
& \dot{y}_3=-z_2 ; \\
& \frac{d \mathbf{u}}{d t}=\left(\begin{array}{llll}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 2 \\
1 & 0 & 0 & 0 \\
0 & 2 & 0 & 0
\end{array}\right) \mathbf{u} .
\end{aligned}
$$

Nicholas Barvinok
Nicholas Barvinok
Numerade Educator
06:35

Problem 46

State and prove a counterpart to Definition 10.32 and Theorem 10.33 for a homogeneous linear iterative system.

Uma Kumari
Uma Kumari
Numerade Educator
11:39

Problem 47

. Solve the following initial value problems:
(a)
$$
\begin{array}{ll}
\dot{u}_1=2 u_1-u_2, & u_1(0)=0 \\
\dot{u}_2=4 u_1-3 u_2+e^{2 t}, & u_2(0)=0 .
\end{array}
$$
(b) $\dot{u}_1=-u_1+2 u_2+e^t, \quad u_1(1)=1$,
$$
\begin{aligned}
& \dot{u}_2=2 u_1-u_2+e^t, \quad u_2(1)=1 \text {. } \\
& \dot{u}=3 u+v+1, \quad u(1)=1 \text {, } \\
& \dot{v}=4 u+t, \quad v(1)=-1 \text {. } \\
&
\end{aligned}
$$
(c) $\dot{u}_1=-u_2, \quad u_1(0)=0$,
(e) $\begin{array}{ll}\dot{p}=p+q+t, & p(0)=0, \\ \dot{q}=-p-q+t, & q(0)=0 .\end{array}$

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
11:39

Problem 48

Solve the following initial value problems:
$$
\begin{array}{llll}
\dot{u}_1=-2 u_2+2 u_3, & u_1(0)=1, & \dot{u}_1=u_1-2 u_2, & u_1(0)=-1, \\
\dot{u}_2=-u_1+u_2-2 u_3+t, & u_2(0)=0, & \text { (b) } \quad \dot{u}_2=-u_2+e^{-t,} & u_2(0)=0 \\
\dot{u}_3=-3 u_1+u_2-2 u_3+1, & u_3(0)=0, & \dot{u}_3=4 u_1-4 u_2-u_3, & u_3(0)=-1 .
\end{array}
$$
(a) $\dot{u}_2=-u_1+u_2-2 u_3+t, \quad u_2(0)=0$,
(b)

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator

Problem 49

Suppose that $\lambda$ is not an eigenvalue of $A$. Show that the inhomogeneous system $\dot{\mathbf{u}}=A \mathbf{u}+e^{\lambda t} \mathbf{v}$ has a solution of the form $\mathbf{u}^*(t)=e^{\lambda t} \mathbf{w}$, where $\mathbf{w}$ is a constant vector. What is the general solution?

Check back soon!
02:26

Problem 50

(a) Write down an integral formula for the solution to the initial value problem $\frac{d \mathbf{u}}{d t}=A \mathbf{u}+\mathbf{b}, \mathbf{u}(0)=\mathbf{0}$, where $\mathbf{b}$ is a constant vector.
(b) Suppose $\mathbf{b} \in \operatorname{img} A$. Do you recover the solution you found in Exercise 10.2.24?

Zachary Mitchell
Zachary Mitchell
Numerade Educator