Elementary Differential Equations with Boundary Value Problems

Werner E. Kohler, Lee W. Johnson

Chapter 4

First Order Linear Systems - all with Video Answers

Educators

Section 1

Introduction

Problem 1

Exercises 1-5:
For the given matrix functions $A(t), B(t)$, and $\mathbf{c}(t)$, make the indicated calculations
$$
A(t)=\left[\begin{array}{cc}
t-1 & t^{2} \\
2 & 2 t+1
\end{array}\right], \quad B(t)=\left[\begin{array}{cc}
t & -1 \\
0 & t+2
\end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c}
t+1 \\
-1
\end{array}\right]
$$
2 A(t)-3 t B(t)

Tamara Worner

Tamara Worner

Numerade Educator

Problem 2

Exercises 1-5:
For the given matrix functions $A(t), B(t)$, and $\mathbf{c}(t)$, make the indicated calculations
$$
A(t)=\left[\begin{array}{cc}
t-1 & t^{2} \\
2 & 2 t+1
\end{array}\right], \quad B(t)=\left[\begin{array}{cc}
t & -1 \\
0 & t+2
\end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c}
t+1 \\
-1
\end{array}\right]
$$
$$
\mathrm{A}(t) B(t)-B(t) A(t)
$$

Tamara Worner

Numerade Educator

Problem 3

Exercises 1-5:
For the given matrix functions $A(t), B(t)$, and $\mathbf{c}(t)$, make the indicated calculations
$$
A(t)=\left[\begin{array}{cc}
t-1 & t^{2} \\
2 & 2 t+1
\end{array}\right], \quad B(t)=\left[\begin{array}{cc}
t & -1 \\
0 & t+2
\end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c}
t+1 \\
-1
\end{array}\right]
$$
$$
A(t) \mathbf{c}(t)
$$

Tamara Worner

Numerade Educator

Problem 4

Exercises 1-5:
For the given matrix functions $A(t), B(t)$, and $\mathbf{c}(t)$, make the indicated calculations
$$
A(t)=\left[\begin{array}{cc}
t-1 & t^{2} \\
2 & 2 t+1
\end{array}\right], \quad B(t)=\left[\begin{array}{cc}
t & -1 \\
0 & t+2
\end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c}
t+1 \\
-1
\end{array}\right]
$$
$$
\operatorname{det}[t A(t)]
$$

Tamara Worner

Numerade Educator

Problem 5

Exercises 1-5:
For the given matrix functions $A(t), B(t)$, and $\mathbf{c}(t)$, make the indicated calculations
$$
A(t)=\left[\begin{array}{cc}
t-1 & t^{2} \\
2 & 2 t+1
\end{array}\right], \quad B(t)=\left[\begin{array}{cc}
t & -1 \\
0 & t+2
\end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c}
t+1 \\
-1
\end{array}\right]
$$
$$
\operatorname{det}[B(t) A(t)]
$$

Tamara Worner

Numerade Educator

Problem 6

Determine all values $t$ such that $A(t)$ is invertible and, for those $t$-values, find $A^{-1}(t)$
$$
A(t)=\left|\begin{array}{cc}
t+1 & t \\
t & t+1
\end{array}\right|
$$

ET

Ed Tam

Numerade Educator

Problem 7

Determine all values $t$ such that $A(t)$ is invertible and, for those $t$-values, find $A^{-1}(t)$
$$
A(t)=\left[\begin{array}{cc}
t & 2 \\
2 & t-3
\end{array}\right]
$$

Michael Jacobsen

Michael Jacobsen

Numerade Educator

Problem 8

Determine all values $t$ such that $A(t)$ is invertible and, for those $t$-values, find $A^{-1}(t)$
$$
\text { 3. } A(t)=\left[\begin{array}{lr}
\sin t & -\cos t \\
\sin t & \cos t
\end{array}\right]
$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 9

Determine all values $t$ such that $A(t)$ is invertible and, for those $t$-values, find $A^{-1}(t)$
$$
A(t)=\left[\begin{array}{ll}
e^{t} & e^{3 t} \\
e^{2 t} & e^{4 t}
\end{array}\right]
$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 10

Find $\lim _{t \rightarrow 0} A(t)$ or state that the limit does not exist.
$$
A(t)=\left[\begin{array}{ccc}
\frac{t}{t} & t \cos t & \overline{t+1} \\
e^{3 t} & \sec t & \frac{2 t}{t^{2}-1}
\end{array}\right]
$$

Carson Merrill

Numerade Educator

Problem 11

Find $\lim _{t \rightarrow 0} A(t)$ or state that the limit does not exist.
$$
A(t)=\left[\begin{array}{cc}
t e^{-t} & \tan t \\
t^{2}-2 & e^{\sin t}
\end{array}\right]
$$

Carson Merrill

Numerade Educator

Problem 12

Find $A^{\prime}(t)$ and $A^{\prime \prime}(t)$. For what values of $t$ are the matrices $A(t), A^{\prime}(t)$, and $A^{\prime \prime}(t)$ defined?
$$
A(t)=\left[\begin{array}{cc}
\sin t & 3 t \\
t^{2}+2 & 5
\end{array}\right]
$$

Aman Gupta

Numerade Educator

Problem 13

Find $A^{\prime}(t)$ and $A^{\prime \prime}(t)$. For what values of $t$ are the matrices $A(t), A^{\prime}(t)$, and $A^{\prime \prime}(t)$ defined?
$$
A(t)=\left[\begin{array}{cc}
7 & \ln |t| \\
\sqrt{1-t} & e^{3 t}
\end{array}\right]
$$

Bobby Barnes

University of North Texas

Problem 14

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
\begin{aligned}
&y_{1}^{\prime}=t^{2} y_{1}+3 y_{2}+\sec t \\
&y_{2}^{\prime}=(\sin t) y_{1}+t y_{2}-5
\end{aligned}
$$

Christian Otero

Christian Otero

Numerade Educator

Problem 15

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
\begin{aligned}
&y_{1}^{\prime}=t^{-1} y_{1}+\left(t^{2}+1\right) y_{2}+t \\
&y_{2}^{\prime}=4 y_{1}+t^{-1} y_{2}+8 t \ln t
\end{aligned}
$$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 16

Determine the general form of $A(t)$ by constructing antiderivatives as needed and imposing any given constraints.$$
A^{\prime}(t)=\left[\begin{array}{c}
-1 \\
2 t
\end{array}\right]
$$

Amrita Bhasin

Numerade Educator

Problem 17

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime}(t)=\left[\begin{array}{lll}
1 & t & e^{t}
\end{array}\right], \quad A(0)=\left[\begin{array}{lll}
-1 & 1 & 0
\end{array}\right]
$$

Tyler Moulton

Numerade Educator

Problem 18

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime}(t)=\left[\begin{array}{cc}
2 t & 1 \\
\cos t & 3 t^{2}
\end{array}\right], \quad A(0)=\left[\begin{array}{rr}
2 & 5 \\
1 & -2
\end{array}\right]
$$

Christian Otero

Christian Otero

Numerade Educator

Problem 19

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime}(t)=\left[\begin{array}{cc}
t^{-1} & 4 t \\
5 & 3 t^{2}
\end{array}\right], \quad A(1)=\left[\begin{array}{rr}
2 & 5 \\
1 & -2
\end{array}\right]
$$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 20

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime \prime}(t)=\left[\begin{array}{l}
2 \\
e^{t}
\end{array}\right]
$$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 21

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime \prime}(t)=\left[\begin{array}{cc}
2 t & \sin t \\
0 & 0
\end{array}\right], \quad A(0)=\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right], \quad A^{\prime}(0)=\left[\begin{array}{ll}
0 & 0 \\
1 & 0
\end{array}\right]
$$

Tyler Moulton

Numerade Educator

Problem 22

Each of the systems of linear differential equations can be expressed in the form $\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .$ Determine $P(t)$ and $\mathbf{g}(t)$
$$
A^{\prime \prime}(t)=\left[\begin{array}{ll}
1 & t \\
0 & 0
\end{array}\right], \quad A(0)=\left[\begin{array}{rr}
1 & 1 \\
-2 & 1
\end{array}\right], \quad A(1)=\left[\begin{array}{ll}
-1 & 2 \\
-2 & 3
\end{array}\right]
$$

Brian Ketelobeter

Brian Ketelobeter

Numerade Educator

Problem 23

Calculate $A(t)=\int_{0}^{t} B(s) d s$.
$$
B(s)=\left[\begin{array}{ccc}
2 s & \cos s & 2 \\
5 & (s+1)^{-1} & 3 s^{2}
\end{array}\right]
$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 24

Calculate $A(t)=\int_{0}^{t} B(s) d s$.
$$
B(s)=\left[\begin{array}{cc}
e^{s} & 6 s \\
\cos 2 \pi s & \sin 2 \pi s
\end{array}\right]
$$

Nick Johnson

Numerade Educator

Problem 25

Let $A(t)$ be an $(n \times n)$ matrix function. We use the notation $A^{2}(t)$ to mean the matrix function $A(t) A(t)$.
(a) Construct an explicit $(2 \times 2)$ differentiable matrix function to show that $\frac{d}{d t}\left[A^{2}(t)\right] \quad$ and $\quad 2 A(t) \frac{d}{d t}[A(t)]$ are generally not equal.(b) What is the correct formula relating the derivative of $A^{2}(t)$ to the matrices $A(t)$ and $A^{\prime}(t)$ ?

Brandon Collins

Brandon Collins

Numerade Educator

Problem 26

Construct an example of a $(2 \times 2)$ matrix function $A(t)$ such that $A^{2}(t)$ is a constant matrix but $A(t)$ is not a constant matrix.

Chandra Jain

Numerade Educator

Problem 27

Let $A(t)$ be an ( $n \times n$ ) matrix function that is both differentiable and invertible on some $t$-interval of interest. It can be shown that $A^{-1}(t)$ is likewise differentiable on this interval. Differentiate the matrix identity $A^{-1}(t) A(t)=I$ to obtain the following formula:
$$
\frac{d}{d t}\left[A^{-1}(t)\right]=-A^{-1}(t) A^{\prime}(t) A^{-1}(t)
$$
[Hint: Recall the product rule, equation (9). Notice that the formula you derive is not the same as the power rule of single-variable calculus.]

Nick Johnson

Numerade Educator

Problem 28

Consider the matrix function
$$
A(t)=\left[\begin{array}{ll}
t & t^{3} \\
0 & 2 t
\end{array}\right]
$$
Explicitly calculate both $(d / d t)\left[A^{-1}(t)\right]$ and $-A^{-1}(t) A^{\prime}(t) A^{-1}(t)$ for this special case to illustrate the formula derived in Exercise $27 .$

Elham Kordzadeh

Elham Kordzadeh

Numerade Educator

Problem 29

Consider the two-tank mixing apparatus shown in the figure. Each tank has a capacity of 500 gal and initially contains 100 gal of fresh water. At time $t=0$, the well-stirred mixing process begins with the specified input concentration and flow rates.
Figure for Exercises 29-32
(a) Determine the volume of solution in each tank as a function of time.
(b) Determine the time interval of interest. (The process stops when a tank is full or empty.)
(c) Let $Q_{1}(t)$ and $Q_{2}(t)$ denote the amounts of salt (in pounds) in Tanks 1 and 2 at time $t$ (in minutes). Derive the initial value problem with $Q_{1}$ and $Q_{2}$ as dependent variables describing the mixing process.
$$
r_{1}=r_{3}=5 \mathrm{gal} / \mathrm{min}, \quad r_{2}=r_{4}=10 \mathrm{gal} / \mathrm{min}
$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 30

Consider the two-tank mixing apparatus shown in the figure. Each tank has a capacity of 500 gal and initially contains 100 gal of fresh water. At time $t=0$, the well-stirred mixing process begins with the specified input concentration and flow rates.
Figure for Exercises 29-32
(a) Determine the volume of solution in each tank as a function of time.
(b) Determine the time interval of interest. (The process stops when a tank is full or empty.)
(c) Let $Q_{1}(t)$ and $Q_{2}(t)$ denote the amounts of salt (in pounds) in Tanks 1 and 2 at time $t$ (in minutes). Derive the initial value problem with $Q_{1}$ and $Q_{2}$ as dependent variables describing the mixing process.
$$
r_{1}=r_{3}=5 \mathrm{gal} / \mathrm{min}, \quad r_{2}=6 \mathrm{gal} / \mathrm{min}, \quad r_{4}=4 \mathrm{gal} / \mathrm{min}
$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 31

Consider the two-tank mixing apparatus shown in the figure. Each tank has a capacity of 500 gal and initially contains 100 gal of fresh water. At time $t=0$, the well-stirred mixing process begins with the specified input concentration and flow rates.
Figure for Exercises 29-32
(a) Determine the volume of solution in each tank as a function of time.
(b) Determine the time interval of interest. (The process stops when a tank is full or empty.)
(c) Let $Q_{1}(t)$ and $Q_{2}(t)$ denote the amounts of salt (in pounds) in Tanks 1 and 2 at time $t$ (in minutes). Derive the initial value problem with $Q_{1}$ and $Q_{2}$ as dependent variables describing the mixing process.
$$
r_{1}=5 \mathrm{gal} / \mathrm{min}, \quad r_{3}=0, \quad r_{2}=r_{4}=5 \mathrm{gal} / \mathrm{min}
$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 32

Consider the two-tank mixing apparatus shown in the figure. Each tank has a capacity of 500 gal and initially contains 100 gal of fresh water. At time $t=0$, the well-stirred mixing process begins with the specified input concentration and flow rates.
Figure for Exercises 29-32
(a) Determine the volume of solution in each tank as a function of time.
(b) Determine the time interval of interest. (The process stops when a tank is full or empty.)
(c) Let $Q_{1}(t)$ and $Q_{2}(t)$ denote the amounts of salt (in pounds) in Tanks 1 and 2 at time $t$ (in minutes). Derive the initial value problem with $Q_{1}$ and $Q_{2}$ as dependent variables describing the mixing process.
$$
r_{1}=5 \mathrm{gal} / \mathrm{min}, \quad r_{3}=10 \mathrm{gal} / \mathrm{min}, \quad r_{2}=r_{4}=5 \mathrm{gal} / \mathrm{min}
$$

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator