Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 5

Eigenvalues and Eigenvectors - all with Video Answers

Educators

Section 7

Applications to Differential Equations

Problem 1

A particle moving in a planar force field has a position vector $x$ that satisfies $\mathbf{x}^{\prime}=A \mathbf{x}$ . The $2 \times 2$ matrix $A$ has eigenvalues 4 and $2,$ with corresponding eigenvectors $\mathbf{v}_{1}=\left[\begin{array}{r}{-3} \\ {1}\end{array}\right]$ and $\mathbf{v}_{2}=\left[\begin{array}{r}{-1} \\ {1}\end{array}\right] .$ Find the position of the particle at time $t$ assuming that $\mathbf{x}(0)=\left[\begin{array}{r}{-6} \\ {1}\end{array}\right]$

Ibrahima B.

Ibrahima B.

Numerade Educator

Problem 2

Let $A$ be a $2 \times 2$ matrix with eigenvalues $-3$ and $-1$ and corresponding eigenvectors $\mathbf{v}_{1}=\left[\begin{array}{r}{-1} \\ {1}\end{array}\right]$ and $\mathbf{v}_{2}=\left[\begin{array}{l}{1} \\ {1}\end{array}\right] .$ Let $\mathbf{x}(t)$ be the position of a particle at time $t .$ Solve the initial value problem $\mathbf{x}^{\prime}=A \mathbf{x}, \mathbf{x}(0)=\left[\begin{array}{l}{2} \\ {3}\end{array}\right]$

James C.

Numerade Educator

Problem 3

In Exercises $3-6,$ solve the initial value problem $\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$ for $t \geq 0,$ with $\mathbf{x}(0)=(3,2) .$ Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by $\mathbf{x}^{\prime}=A \mathbf{x}$ . Find the directions of greatest attraction and/or repulsion. When the origin is a saddle point, sketch typical trajectories.
$$
A=\left[\begin{array}{rr}{2} & {3} \\ {-1} & {-2}\end{array}\right]
$$

Bobby B.

University of North Texas

Problem 4

In Exercises $3-6,$ solve the initial value problem $\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$ for $t \geq 0,$ with $\mathbf{x}(0)=(3,2) .$ Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by $\mathbf{x}^{\prime}=A \mathbf{x}$ . Find the directions of greatest attraction
and/or repulsion. When the origin is a saddle point, sketch typical traiectories.
$$
A=\left[\begin{array}{rr}{-2} & {-5} \\ {1} & {4}\end{array}\right]
$$

James C.

Numerade Educator

Problem 5

In Exercises $3-6,$ solve the initial value problem $\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$ for $t \geq 0,$ with $\mathbf{x}(0)=(3,2) .$ Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by $\mathbf{x}^{\prime}=A \mathbf{x}$ . Find the directions of greatest attraction
and/or repulsion. When the origin is a saddle point, sketch typical traiectories.
$$
A=\left[\begin{array}{rr}{7} & {-1} \\ {3} & {3}\end{array}\right]
$$

Check back soon!

Problem 6

In Exercises $3-6,$ solve the initial value problem $\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$ for $t \geq 0,$ with $\mathbf{x}(0)=(3,2) .$ Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by $\mathbf{x}^{\prime}=A \mathbf{x}$ . Find the directions of greatest attraction
and/or repulsion. When the origin is a saddle point, sketch typical traiectories.
$$
A=\left[\begin{array}{ll}{1} & {-2} \\ {3} & {-4}\end{array}\right]
$$

James C.

Numerade Educator

Problem 7

In Exercises 7 and $8,$ make a change of variable that decouples the equation $\mathbf{x}^{\prime}=A \mathbf{x}$ . Write the equation $\mathbf{x}(t)=P \mathbf{y}(t)$ and show the calculation that leads to the uncoupled system $\mathbf{y}^{\prime}=D \mathbf{y},$ specifying $P$ and $D .$
$A$ as in Exercise 5

Ibrahima B.

Numerade Educator

Problem 8

In Exercises 7 and $8,$ make a change of variable that decouples the equation $\mathbf{x}^{\prime}=A \mathbf{x}$ . Write the equation $\mathbf{x}(t)=P \mathbf{y}(t)$ and show the calculation that leads to the uncoupled system $\mathbf{y}^{\prime}=D \mathbf{y},$ specifying $P$ and $D .$
$A$ as in Exercise 6

James C.

Numerade Educator

Problem 9

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{ll}{-3} & {2} \\ {-1} & {-1}\end{array}\right]
$$

Ibrahima B.

Numerade Educator

Problem 10

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{rl}{3} & {1} \\ {-2} & {1}\end{array}\right]
$$

Ibrahima B.

Numerade Educator

Problem 11

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{rr}{-3} & {-9} \\ {2} & {3}\end{array}\right]
$$

Ibrahima B.

Numerade Educator

Problem 12

In Exercises $9-18,$ construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigen functions and then obtain the general real solution. Describe the shapes of typical trajectories. $A=\left[\begin{array}{rr}-7 & 10 \\ -4 & 5\end{array}\right]$

Bobby B.

University of North Texas

Problem 13

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{ll}{4} & {-3} \\ {6} & {-2}\end{array}\right]
$$

Check back soon!

Problem 14

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{ll}{-2} & {1} \\ {-8} & {2}\end{array}\right]
$$

Bobby B.

University of North Texas

Problem 15

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
[\mathbf{M}] A=\left[\begin{array}{rrr}{-8} & {-12} & {-6} \\ {2} & {1} & {2} \\ {7} & {12} & {5}\end{array}\right]
$$

Check back soon!

Problem 16

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
[\mathbf{M}] A=\left[\begin{array}{rrr}{-6} & {-11} & {16} \\ {2} & {5} & {-4} \\ {-4} & {-5} & {10}\end{array}\right]
$$

James C.

Numerade Educator

Problem 17

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
[\mathbf{M}] A=\left[\begin{array}{rrr}{30} & {64} & {23} \\ {-11} & {-23} & {-9} \\ {6} & {15} & {4}\end{array}\right]
$$

Check back soon!

Problem 18

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
[\mathbf{M}] A=\left[\begin{array}{rrr}{53} & {-30} & {-2} \\ {90} & {-52} & {-3} \\ {20} & {-10} & {2}\end{array}\right]
$$

James C.

Numerade Educator

Problem 19

[M] Find formulas for the voltages $v_{1}$ and $v_{2}$ (as functions of time $t )$ for the circuit in Example $1,$ assuming that $R_{1}=1 / 5$ ohm, $R_{2}=1 / 3$ ohm, $C_{1}=4$ farads, $C_{2}=3$ farads, and the initial charge on each capacitor is 4 volts.

Check back soon!

Problem 20

[M] Find formulas for the voltages $v_{1}$ and $v_{2}$ for the circuit in Example $1,$ assuming that $R_{1}=1 / 15$ ohm, $R_{2}=1 / 3$ ohm, $C_{1}=9$ farads $, C_{2}=2$ farads, and the initial charge on each capacitor is 3 volts.

James C.

Numerade Educator

Problem 21

IM Find formulas for the current $i_{L}$ and the voltage $v_{C}$ for the circuit in Example $3,$ assuming that $R_{1}=1$ ohm, $R_{2}=.125$ ohm, $C=.2$ farad, $L=.125$ henry, the initial current is 0 amp, and the initial voltage is 15 volts.

Check back soon!

Problem 22

IMI The circuit in the figure is described by the equation
$$
\left[\begin{array}{c}{i_{L}^{\prime}} \\ {v_{c}^{\prime}}\end{array}\right]=\left[\begin{array}{cc}{0} & {1 / L} \\ {-1 / C} & {-1 /(R C)}\end{array}\right]\left[\begin{array}{c}{i_{L}} \\ {v_{c}}\end{array}\right]
$$
where $i_{L}$ is the current through the inductor $L$ and $v_{C}$ is the voltage drop across the capacitor $C .$ Find formulas for $i_{L}$ and $v_{C}$ when $R=.5$ ohm, $C=2.5$ farads, $L=.5$ henry,
the initial current is 0 amp, and the initial voltage is 12 volts.

James C.

Numerade Educator