• Home
  • Textbooks
  • Engineering Differential Equations: Theory and Applications
  • Systems of First-Order Linear Constant-Coefficient Ordinary Differential Equations

Engineering Differential Equations: Theory and Applications

Bill Goodwine

Chapter 6

Systems of First-Order Linear Constant-Coefficient Ordinary Differential Equations - all with Video Answers

Educators

Chapter Questions

04:58

Problem 1

Each of the matrices in this problem has a full set of linearly independent eigenvectors. For each one, indicate whether Theorem 6.1 or 6.2 applies and find the general solution to $\dot{\xi}=A \xi$ for:
$$
\begin{array}{lll}
A_1=\left[\begin{array}{cc}
12 & -3 \\
8 & 2
\end{array}\right], & A_2=\left[\begin{array}{cc}
12 & -3 \\
8 & 1
\end{array}\right], & A_3=\left[\begin{array}{ll}
13 & -6 \\
16 & -7
\end{array}\right], \\
A_4=\left[\begin{array}{cc}
7 & -5 \\
2 & 0
\end{array}\right], & A_5=\left[\begin{array}{cc}
-7 & 10 \\
-4 & 7
\end{array}\right], & A_6=\left[\begin{array}{ll}
14 & -5 \\
40 & -16
\end{array}\right]
\end{array}
$$

Christian Otero
Christian Otero
Numerade Educator

Problem 2

Each of the matrices in this problem has a full set of linearly independent eigenvectors. For each one, indicate whether Theorem 6.1 or 6.2 applies and find the general solution to $\dot{\xi}=A \xi$ for:

$$
\begin{aligned}
& A_1=\left[\begin{array}{cc}
6 & -4 \\
0 & 2
\end{array}\right] \text {, } \\
& A_2=\left[\begin{array}{ccc}
-3 & 0 & 0 \\
0 & -3 & 1 \\
0 & 1 & -3
\end{array}\right], \\
& A_3=\left[\begin{array}{ccc}
-3 & 0 & 0 \\
-1 & -3 & 1 \\
-1 & 1 & -3
\end{array}\right], \\
& A_4=\left[\begin{array}{ccc}
-8 & 7 & 1 \\
0 & -1 & 1 \\
0 & 0 & 0
\end{array}\right], \\
& A_5=\left[\begin{array}{llll}
3 & 2 & 0 & 0 \\
2 & 3 & 0 & 0 \\
0 & 0 & 1 & 4 \\
0 & 0 & 4 & 1
\end{array}\right] \text {, } \\
& A_6=\left[\begin{array}{cccc}
-2 & 0 & 0 & 0 \\
0 & -2 & 0 & 0 \\
0 & 0 & 0 & 2 \\
0 & 0 & 2 & 0
\end{array}\right], \\
& A_7=\left[\begin{array}{cccc}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 9 \\
0 & 2 & 1 & 4 \\
0 & -4 & 0 & 14
\end{array}\right], \quad A_8=\left[\begin{array}{ccc}
-3 & 1 & 0 \\
0 & -2 & 0 \\
1 & 1 & -4
\end{array}\right], \quad A_9=\left[\begin{array}{ccc}
2 & 0 & 3 \\
0 & -5 & 0 \\
3 & 0 & 2
\end{array}\right] .
\end{aligned}
$$

Check back soon!

Problem 3

For $A_2, A_3, A_4, A_8$, and $A_9$ in Exercise 6.2 , determine the solution if $\xi_1(0)=1$, $\xi_2(0)=2$, and $\xi_3(0)=4$.

Check back soon!
06:11

Problem 4

Each of the matrices in this problem has some complex eigenvalues. Determine the general solution to $\dot{\xi}=A \xi$ for:

$$
\begin{array}{lll}
A_1=\left[\begin{array}{cc}
-1 & 2 \\
-2 & -1
\end{array}\right], & A_2=\left[\begin{array}{cc}
-1 & 1 \\
-10 & 5
\end{array}\right], & A_3=\left[\begin{array}{ll}
-12 & 10 \\
-20 & 16
\end{array}\right], \\
A_4=\left[\begin{array}{ll}
8 & -10 \\
4 & -4
\end{array}\right], & A_5=\left[\begin{array}{cc}
4 & 4 \\
-2 & 0
\end{array}\right], & A_6=\left[\begin{array}{lc}
-2 & 2 \\
-1 & -4
\end{array}\right] .
\end{array}
$$

Ryan Williams
Ryan Williams
Numerade Educator

Problem 5

Each of the matrices in this problem has some complex eigenvalues. Determine the general solution to $\dot{\xi}=A \xi$ for:
$$
\begin{array}{ll}
A_1=\left[\begin{array}{cccc}
0 & 1 & 0 & 0 \\
-4 & 4 & 0 & 0 \\
0 & 0 & 3 & 2 \\
0 & 0 & -2 & 3
\end{array}\right], & A_2=\left[\begin{array}{ccc}
-\frac{7}{2} & \frac{15}{2} & -3 \\
-\frac{3}{2} & -\frac{1}{2} & 3 \\
0 & 0 & 1
\end{array}\right],
\end{array} \begin{array}{ll}
A_3=\left[\begin{array}{cc}
-1 & -4 \\
4 & -1
\end{array}\right], \\
A_4=\left[\begin{array}{ccc}
11 & 0 & 17 \\
0 & -6 & 0 \\
-2 & 0 & 1
\end{array}\right], & A_5=\left[\begin{array}{cccc}
-5 & 1 & 0 & 0 \\
-1 & -3 & 0 & 0 \\
0 & 0 & -1 & -4 \\
0 & 0 & 2 & -5
\end{array}\right],
\end{array} \begin{array}{|ll}
A_6 & =\left[\begin{array}{ccccc}
-5 & 0 & 0 & 0 \\
0 & -3 & 0 & 0 & 0 \\
0 & -4 & 1 & 0 & 0 \\
0 & 0 & 0 & -5 & 1 \\
0 & 0 & 0 & -1 & -7
\end{array}\right], \\
A_7=\left[\begin{array}{ccc}
-3 & 0 & 0 \\
0 & -5 & 6 \\
0 & -3 & 1
\end{array}\right], & A_8=\left[\begin{array}{ccc}
2 & 0 & -3 \\
0 & -5 & 0 \\
3 & 0 & 2
\end{array}\right],
\end{array} \quad A_9=\left[\begin{array}{cccc}
-1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0 \\
0 & 0 & -1 & 2 \\
0 & 0 & -2 & -1
\end{array}\right] .
$$

Check back soon!

Problem 6

For $A_2, A_4, A_7$, and $A_8$ in Exercise 6.5, determine the solution if $\xi_1(0)=1$, $\xi_2(0)=1$, and $\xi_3(0)=0$.

Check back soon!
06:11

Problem 7

Each of the matrices in this problem has some repeated eigenvalues. Determine the general solution to $\dot{\xi}=A \xi$ for:

$$
\begin{array}{lll}
A_1=\left[\begin{array}{cc}
-2 & 1 \\
0 & -2
\end{array}\right], & A_2=\left[\begin{array}{cc}
-3 & 1 \\
-1 & -5
\end{array}\right], & A_3=\left[\begin{array}{cc}
-8 & 1 \\
-4 & -4
\end{array}\right], \\
A_4=\left[\begin{array}{cc}
-11 & 1 \\
-9 & -5
\end{array}\right], & A_5=\left[\begin{array}{cc}
-11 & 1 \\
-4 & -7
\end{array}\right], & A_6=\left[\begin{array}{ll}
-7 & 2 \\
-81
\end{array}\right] .
\end{array}
$$

Ryan Williams
Ryan Williams
Numerade Educator

Problem 8

Each of the matrices in this problem has some repeated eigenvalues. Determine the general solution to $\dot{\xi}=A \xi$ for:

$$
\left.\begin{array}{lll}
A_1=\left[\begin{array}{ccc}
-1 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & -2
\end{array}\right], & A_2=\left[\begin{array}{ccc}
-1 & 0 & 0 \\
0 & -2 & 1 \\
0 & 0 & -2
\end{array}\right], & A_3=\left[\begin{array}{ccc}
-2 & 1 & 0 \\
0 & -2 & 0 \\
0 & 1 & -2
\end{array}\right], \\
A_4=\left[\begin{array}{ccc}
6 & 0 & 0 \\
1 & 5 & 1 \\
1 & -1 & 7
\end{array}\right], & A_5=\left[\begin{array}{ccccc}
-4 & 1 & 0 & 0 & 0 \\
0 & -4 & 0 & 0 & 0 \\
0 & 0 & -4 & 0 & 0 \\
0 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & -1
\end{array}\right], A_6=\left[\begin{array}{cccc}
-4 & 1 & 0 & 0 \\
0 & -4 & 1 & 0 \\
0 \\
0 & 0 & -4 & 0 \\
0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0
\end{array}\right] .
\end{array}\right] .
$$

Check back soon!
04:56

Problem 9

Prove Theorem 6.4 by substituting Equation (6.24) into $\dot{\xi}=A \xi$ and making use of the properties of generalized eigenvectors.

Ryan Williams
Ryan Williams
Numerade Educator

Problem 10

. Find the general solution to

$$
\frac{\mathrm{d}}{\mathrm{~d} t}\left[\begin{array}{l}
\xi_1 \\
\xi_2
\end{array}\right]=\left[\begin{array}{ll}
2 & 1 \\
0 & 3
\end{array}\right]\left[\begin{array}{l}
\xi_1 \\
\xi_2
\end{array}\right]+\left[\begin{array}{c}
0 \\
\mathrm{e}^{-t}
\end{array}\right]
$$

Check back soon!

Problem 11

Consider
$$
\frac{\mathrm{d}}{\mathrm{~d} t}\left[\begin{array}{l}
\xi_1 \\
\xi_2
\end{array}\right]=\left[\begin{array}{ll}
2 & 1 \\
0 & 3
\end{array}\right]\left[\begin{array}{l}
\xi_1 \\
\xi_2
\end{array}\right]+\left[\begin{array}{c}
0 \\
\mathrm{e}^{3 t}
\end{array}\right]
$$

- What happens when you assume

$$
\xi_p(t)=a \mathrm{e}^{3 t}=\left[\begin{array}{l}
a_1 \\
a_2
\end{array}\right] \mathrm{e}^{3 t ?}
$$

Explain why it does not work.
- What happens when you assume

$$
\xi_p(t)=a t \mathrm{e}^{3 t}=t\left[\begin{array}{l}
a_1 \\
a_2
\end{array}\right] \mathrm{e}^{3 t ?}
$$

Explain why it does not work.

Check back soon!

Problem 12

Determine the solution to $\dot{\xi}=A \xi+g(t)$ where

$$
A=\left[\begin{array}{ccc}
-3 & 1 & 0 \\
0 & -2 & 0 \\
1 & 1 & -4
\end{array}\right], \quad g(t)=\left[\begin{array}{c}
0 \\
0 \\
\cos t
\end{array}\right]
$$

- Using the method of undetermined coefficients
- By determining a coordinate transformation that diagonalizes $A$
- Using the method of variation of parameters

Check back soon!
View

Problem 13

Determine the solution to $\dot{\xi}=A \xi+g(t)$ where

$$
A=\left[\begin{array}{ccc}
-3 & 0 & 1 \\
0 & -2 & 0 \\
1 & 0 & -3
\end{array}\right], \quad g(t)=\left[\begin{array}{c}
\mathrm{e}^{-4} \\
0 \\
0
\end{array}\right]
$$

- Using the method of undetermined coefficients
- By determining a coordinate transformation that diagonalizes $A$
- Using the method of variation of parameters.

Because $A=A^T$, make use of the fact that $T^{-1}=T^T$ (as long as you normalize all the eigenvectors to have unit length). Verify this fact by showing that $T^T T=I$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 14

. Compute the matrix exponential for $A_1, A_2$, and $A_9$ from Exercise 6.2. For the initial condition given in Exercise $6.3, A_3$ and $A_9$ verify that

$$
\xi(t)=\mathrm{e}^{A t} \xi(0)
$$

is the same solution as was computed in Exercise 6.3.

Check back soon!

Problem 15

When an $n \times n$ matrix does not have a linearly independent set of $n$ eigenvectors, the matrix cannot be diagonalized. However, it is still possible to convert it into a simpler and useful form called Jordan canonical form. For the matrices $A_3$ and $A_4$ from Exercise 6.8, construct the matrix $T$ from the generalized eigenvectors and compute $T^{-1} A T$. Explain the manner in which the resulting matrix would be useful to solve either a homogeneous or inhomogeneous set of differential equations containing $A$.

Check back soon!

Problem 16

Determine the general solution to

$$
\frac{\mathrm{d}}{\mathrm{~d} t}\left[\begin{array}{l}
\xi_1 \\
\xi_2 \\
\xi_3
\end{array}\right]=\left[\begin{array}{ccc}
-6 & 0 & 0 \\
1 & -5 & 1 \\
1 & -1 & -7
\end{array}\right]\left[\begin{array}{l}
\xi_1 \\
\xi_2 \\
\xi_3
\end{array}\right]+\left[\begin{array}{l}
0 \\
t \\
0
\end{array}\right] .
$$

Plot the solutions. Write a computer program to determine an approximate numerical solution and compare the answers.

Check back soon!
02:55

Problem 17

Determine the solution to

$$
\begin{aligned}
\ddot{x}+\dot{x}+x-y & =0 \\
\dot{y}-x+2 y & =\mathrm{e}^{-t},
\end{aligned}
$$

where $x(0)=1, \dot{x}(0)=-1$ and $y(0)=2$.

Jack Chen
Jack Chen
Numerade Educator
05:07

Problem 19

Prove that the principle of superposition holds for systems of linear first-order ordinary differential equations; that is, if $\xi_1(t)$ and $\xi_2(t)$ both satisfy $\dot{\xi}=A \xi$, then $\xi(t)=c_1 \xi_1(t)+c_2 \xi_2(t)$ also satisfies it.

Amy Jiang
Amy Jiang
Numerade Educator