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Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 8

Differential Equations - all with Video Answers

Educators

Section 1

Basic Ideas

00:11

Problem 1

What is the order of $y^{n}(t)+9 y(t)=10 ?$

Ernest Castorena
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00:09

Problem 2

Is $y^{\prime \prime}(t)+9 y(t)=10$ linear or nonlinear?

Ernest Castorena
Ernest Castorena
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00:09

Problem 3

How many arbitrary constants appear in the general solution of $y^{\prime \prime}(t)+9 y(t)=10 ?$

Ernest Castorena
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00:39

Problem 4

If the general solution of a differential equation is $y(t)=C e^{-3 t}+10,$ what is the solution that satisfies the initial condition $y(0)=5 ?$

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00:32

Problem 5

Does the function $y(t)=2 t$ satisfy the differential equation $y^{-1}(t)+y^{\prime}(t)=2 ?$

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00:43

Problem 6

Does the function $y(t)=6 e^{-3 t}$ satisfy the initial value problem $y^{\prime}(t)-3 y(t)=0, y(0)=6 ?$

Ernest Castorena
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00:24

Problem 7

Verify that the given function y is a solution of the differential equation that follows it. Assume that $C$ is an arbitrary constant.
$$y(t)=C e^{-5 t_{i}}, y^{\prime}(t)+5 y(t)=0$$

Ernest Castorena
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00:35

Problem 8

Verify that the given function y is a solution of the differential equation that follows it. Assume that $C$ is an arbitrary constant.
$$y(t)=C t^{-3} ; t y^{\prime}(t)+3 y(t)=0$$

Ernest Castorena
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00:52

Problem 9

Verify that the given function y is a solution of the differential equation that follows it. Assume that $C$ is an arbitrary constant.
$$y(t)=C_{1} \sin 4 t+C_{2} \cos 4 t ; y^{\prime \prime}(t)+16 y(t)=0$$

Ernest Castorena
Ernest Castorena
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00:27

Problem 10

Verify that the given function y is a solution of the differential equation that follows it. Assume that $C$ is an arbitrary constant.
$$y(x)=C_{1} e^{-x}+C_{2} e^{x} ; y^{m}(x)-y(x)=0$$

Ernest Castorena
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00:38

Problem 11

Verify that the given function $y$ is a solution of the initial value problem that follows it.
$$y(t)=16 e^{2 t}-10 ; y^{\prime}(t)-2 y(t)=20, y(0)=6$$

Ernest Castorena
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00:48

Problem 12

Verify that the given function $y$ is a solution of the initial value problem that follows it.
$$y(t)=8 t^{6}-3 ; t y^{\prime}(t)-6 y(t)=18, y(1)=5$$

Ernest Castorena
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01:31

Problem 13

Verify that the given function $y$ is a solution of the initial value problem that follows it.
$$y(t)=-3 \cos 3 t ; y^{n}(t)+9 y(t)=0, y(0)=-3, y^{\prime}(0)=0$$

Ernest Castorena
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01:20

Problem 14

Verify that the given function $y$ is a solution of the initial value problem that follows it.
$$y(x)=\frac{1}{4}\left(e^{2 x}-e^{-2 x}\right) ; y^{\prime \prime}(x)-4 y(x)=0, y(0)=0, y^{\prime}(0)=1$$

Ernest Castorena
Ernest Castorena
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00:33

Problem 15

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$y^{\prime}(t)=3+e^{-2 t}$$

Ernest Castorena
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00:42

Problem 16

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$y^{\prime}(t)=12 t^{5}-20 t^{4}+2-6 t^{-2}$$

Ernest Castorena
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00:43

Problem 17

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$y^{\prime}(x)=4 \tan 2 x-3 \cos x$$

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00:28

Problem 18

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$p^{\prime}(x)=\frac{16}{x^{9}}-5+14 x^{6}$$

Ernest Castorena
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00:49

Problem 19

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$y^{\prime \prime}(t)=60 t^{4}-4+12 t^{3}$$

Ernest Castorena
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00:51

Problem 20

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$y^{\prime \prime}(t)=15 e^{3 t}+\sin 4 t$$

Ernest Castorena
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01:12

Problem 21

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$u^{\prime \prime}(x)=55 x^{9}+36 x^{7}-21 x^{5}+10 x^{-3}$$

Ernest Castorena
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00:51

Problem 22

Find the general solution of each differential equation. Use $\boldsymbol{C}, \boldsymbol{C}_{1}, \boldsymbol{C}_{2}, \ldots,$ to denote arbitrary constants.
$$v^{\prime \prime}(x)=x e^{x}$$

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00:36

Problem 23

Solve the following initial value problems.
$$y^{\prime}(t)=1+e^{t}, y(0)=4$$

Ernest Castorena
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00:40

Problem 24

Solve the following initial value problems.
$$y^{\prime}(t)=\sin t+\cos 2 t, y(0)=4$$

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01:21

Problem 25

Solve the following initial value problems.
$$y^{\prime}(x)=3 x^{2}-3 x^{-4}, y(1)=0$$

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00:22

Problem 26

Solve the following initial value problems.
$$y^{\prime}(x)=4 \sec ^{2} 2 x, y(0)=8$$

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01:00

Problem 27

Solve the following initial value problems.
$$y^{\prime \prime}(t)=12 t-20 t^{3}, y(0)=1, y^{\prime}(0)=0$$

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00:33

Problem 28

Solve the following initial value problems.
$$u^{\prime \prime}(x)=4 e^{2 x}-8 e^{-2 x}, u(0)=1, u^{\prime}(0)=3$$

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03:03

Problem 29

An object is fired vertically upward with an initial velocity $v(0)=v_{0}$ from an initial position $\bar{s}(0)=s_{0}$.
a. For the following values of $v_{0}$ and $s_{0 .}$ find the position and velocity fiunctions for all times at which the object is above the ground.
b. Find the time at which the highest point of the trajectory is reached and the height of the object at that time.
$$v_{0}=29.4 \mathrm{m} / \mathrm{s}, s_{0}=30 \mathrm{m}$$

Carson Merrill
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01:34

Problem 30

An object is fired vertically upward with an initial velocity $v(0)=v_{0}$ from an initial position $\bar{s}(0)=s_{0}$.
a. For the following values of $v_{0}$ and $s_{0 .}$ find the position and velocity fiunctions for all times at which the object is above the ground.
b. Find the time at which the highest point of the trajectory is reached and the height of the object at that time.
$$v_{0}=49 \mathrm{m} / \mathrm{s}, s_{0}=60 \mathrm{m}$$

Ernest Castorena
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03:07

Problem 31

Consider the harvesting problem in Example 5.
If $r=0.05$ and $p_{0}=1500,$ for what values of $H$ is the amount of the resource increasing? For what value of $H$ is the amount of the resource constant? If $H=100,$ when does the resource vanish?

Ernest Castorena
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01:58

Problem 32

Consider the harvesting problem in Example 5.
If $r=0.05$ and $H=500 .$ for what values of $p_{0}$ is the amount of the resource decreasing? For what value of $p_{0}$ is the amount of the resource constant? If $p_{0}=9000$, when does the resource vanish?

Ernest Castorena
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03:56

Problem 33

Consider the tank problem in Example 6. For the following parameter values, find the water height function. Then determine the approximate time at which the tank is first empty and graph the solution.
$$H=1.96 \mathrm{m}, A=1.5 \mathrm{m}^{2}, a=0.3 \mathrm{m}^{2}$$

Vipender Yadav
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03:32

Problem 34

Consider the tank problem in Example 6. For the following parameter values, find the water height function. Then determine the approximate time at which the tank is first empty and graph the solution.
$$H=2.25 \mathrm{m}, A=2 \mathrm{m}^{2}, a=0.5 \mathrm{m}^{2}$$

Vipender Yadav
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01:26

Problem 35

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The general solution of the differential equation $y^{\prime}(t)=1$ is $y(t)=t$
b. The differential equation $y^{\prime \prime}(t)-y(t) y^{\prime}(t)=0$ is second order and linear.
c. To find the solution of an initial value problem, you usually begin by finding a general solution of the differential equation.

Ernest Castorena
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01:00

Problem 36

Find the general solution of the following differential equations.
$$y^{\prime}(t)=t \ln t+1$$

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00:52

Problem 37

Find the general solution of the following differential equations.
$$u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}$$

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02:11

Problem 38

Find the general solution of the following differential equations.
$$v^{\prime}(t)=\frac{4}{t^{2}-4}$$

Ernest Castorena
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01:17

Problem 39

Find the general solution of the following differential equations.
$$y^{\prime \prime}(x)=\frac{x}{\left(1-x^{2}\right)^{3 / 2}}$$

Ernest Castorena
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00:40

Problem 40

Find the solution of the following initial value problems.
$$y^{\prime}(t)=t e^{\prime}, y(0)=-1$$

Ernest Castorena
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00:45

Problem 41

Find the solution of the following initial value problems.
$$u^{\prime}(x)=\frac{1}{x^{2}+16}-4, u(0)=2$$

Ernest Castorena
Ernest Castorena
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02:08

Problem 42

Find the solution of the following initial value problems.
$$p^{\prime}(x)=\frac{2}{x^{2}+x}, p(1)=0$$

Ernest Castorena
Ernest Castorena
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01:52

Problem 43

Find the solution of the following initial value problems.
$$y^{\prime \prime}(t)=t e^{t}, y(0)=0, y^{\prime}(0)=1$$

Ernest Castorena
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01:24

Problem 44

Verify that the given function is a solution of the differential equation that follows it.
$$u(t)=C e^{1 /(4 t)} ; u^{\prime}(t)+\frac{1}{t^{5}} u(t)=0$$

Ernest Castorena
Ernest Castorena
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02:42

Problem 45

Verify that the given function is a solution of the differential equation that follows it.
$$u(t)=C_{1} e^{t}+C_{2} t e^{t} ; u^{\prime \prime}(t)-2 u^{\prime}(t)+u(t)=0$$

Ernest Castorena
Ernest Castorena
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03:49

Problem 46

Verify that the given function is a solution of the differential equation that follows it.
$$g(x)=C_{1} e^{-2 x}+C_{2} x e^{-2 x}+2: g^{\prime \prime}(x)+4 g^{\prime}(x)+4 g(x)=8$$

Ernest Castorena
Ernest Castorena
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03:39

Problem 47

Verify that the given function is a solution of the differential equation that follows it.
$$u(t)=C_{1} t^{2}+C_{2} t^{3} ; t^{2} u^{\prime \prime}(t)-4 t u^{\prime}(t)+6 u(t)=0$$

Ernest Castorena
Ernest Castorena
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02:00

Problem 48

Verify that the given function is a solution of the differential equation that follows it.
$$u(t)=C_{1} t^{5}+C_{2} t^{-4}-t^{3} ; t^{2} u^{\pi}(t)-20 u(t)=14 t^{3}$$

Ernest Castorena
Ernest Castorena
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02:00

Problem 48

Verify that the given function is a solution of the differential equation that follows it.
$$u(t)=C_{1} t^{5}+C_{2} t^{-4}-t^{3} ; t^{2} u^{\prime \prime}(t)-20 u(t)=14 t^{3}$$

Ernest Castorena
Ernest Castorena
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03:38

Problem 49

Verify that the given function is a solution of the differential equation that follows it.
$$\begin{aligned}
&z(t)=C_{1} e^{-t}+C_{2} e^{2 t}+C_{3 e}^{-3 t}-e^{t}\\
&z^{\prime \prime \prime}(t)+2 z^{\prime \prime}(t)-5 z^{\prime}(t)-6 z(t)=8 e^{t}
\end{aligned}$$

Ernest Castorena
Ernest Castorena
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08:45

Problem 50

A second-order equation Consider the differential equation $y^{\prime \prime}(t)-k^{2} y(t)=0,$ where $k>0$ is a real number.
a. Verify by substitution that when $k=1$, a solution of the equation is $y(t)=C_{1} e^{t}+C_{2} e^{-t} .$ You may assume that this function is the general solution.
b. Verify by substitution that when $k=2$, the general solution of the equation is $y(t)=C_{1} e^{2 t}+C_{2} e^{-2 t}$
c. Give the general solution of the equation for arbitrary $k>0$ and verify your conjecture.
d. For a positive real number $k$, verify that the general solution of the equation may also be expressed in the form $y(t)=C_{1} \cosh k t+C_{2} \sinh k t,$ where cosh and sinh are
the hyperbolic cosine and hyperbolic sine, respectively (Section 6.10 ).

Vipender Yadav
Vipender Yadav
Numerade Educator
08:45

Problem 50

A second-order equation Consider the differential equation $y^{\prime \prime}(t)-k^{2} y(t)=0,$ where $k>0$ is a real number.
a. Verify by substitution that when $k=1,$ a solution of the equation is $y(t)=C_{1} e^{t}+C_{2} e^{-t} .$ You may assume that this function is the general solution.
b. Verify by substitution that when $k=2,$ the general solution of the equation is $y(t)=C_{1} e^{2 t}+C_{2} e^{-2 t}$
c. Give the general solution of the equation for arbitrary $k>0$ and verify your conjecture.
d. For a positive real number $k$, verify that the general solution of the equation may also be expressed in the form $y(t)=C_{1} \cosh k t+C_{2} \sinh k t,$ where cosh and $\sinh$ are
the hyperbolic cosine and hyperbolic sine, respectively (Section 6.10 ).

Vipender Yadav
Vipender Yadav
Numerade Educator
06:32

Problem 51

Another second-order equation Consider the differential equation $y^{n}(t)+k^{2} y(t)=0,$ where $k$ is a positive real number.
a. Verify by substitution that when $k=1$, a solution of the equation is $y(t)=C_{1} \sin t+C_{2} \cos t .$ You may assume that this function is the general solution.
b. Verify by substitution that when $k=2,$ the general solution of the equation is $y(t)=C_{1} \sin 2 t+C_{2} \cos 2 t$
c. Give the general solution of the equation for arbitrary $k>0$ and verify your conjecture.

Vipender Yadav
Vipender Yadav
Numerade Educator
06:25

Problem 52

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we
present methods for solving these differential equations.
Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation $m^{\prime}(t)+k m(t)=I,$ where $m(t)$ is the mass of the drug in the blood at time $t \geq 0, k$ is a constant that describes the rate at which the drug is absorbed, and $I$ is the infusion rate.
a. Show by substitution that if the initial mass of drug in the blood is zero $(m(0)=0),$ then the solution of the initial value problem is $m(t)=\frac{I}{k}\left(1-e^{-i t}\right)$
b. Graph the solution for $I=10 \mathrm{mg} / \mathrm{hr}$ and $k=0.05 \mathrm{hr}^{-1}$
c. Evaluate $\lim m(t),$ the steady-state drug level, and verify the result using the graph in part (b).

Vipender Yadav
Vipender Yadav
Numerade Educator
04:26

Problem 54

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation $v^{\prime}(t)=g-b v,$ where $v(t)$ is the velocity of the object for $t \geq 0$
$g=9.8 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration due to gravity, and $b>0$ is a constant that involves the mass of the object and the air resistance.
a. Verify by substitution that a solution of the equation, subject to the initial condition $v(0)=0,$ is $v(t)=\frac{g}{b}\left(1-e^{-b t}\right)$
b. Graph the solution with $b=0.1 s^{-1}$
c. Using the graph in part (c), estimate the terminal velocity $\lim _{t \rightarrow \infty} v(t)$

Vipender Yadav
Vipender Yadav
Numerade Educator
07:49

Problem 55

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form $y^{\prime}(t)=-k y^{\prime \prime}(t),$ where $y(t)$ is the concentration of the compound for $t \geq 0, k>0$ is a constant that determines the speed of the reaction, and $n$ is a positive integer called the order of the reaction. Assume that the initial concentration of the compound is $y(0)=y_{0}>0$
a. Consider a first-order reaction $(n=1)$ and show that the solution of the initial value problem is $y(t)=y_{0} e^{-k t}$
b. Consider a second-order reaction $(n=2)$ and show that the solution of the initial value problem is $y(t)=\frac{y_{0}}{y_{0} k t+1}$
c. Let $y_{0}=1$ and $k=0.1 .$ Graph the first-order and secondorder solutions found in parts (a) and (b). Compare the two reactions.

Vipender Yadav
Vipender Yadav
Numerade Educator
06:38

Problem 56

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.
Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let $M(t)$ be the mass of a tumor, for $t \geq 0 .$ The relevant initial value problem is
$$\frac{d M}{d t}=-r M(t) \ln \left(\frac{M(t)}{K}\right), \quad M(0)=M_{0}$$
where $r$ and $K$ are positive constants and $0<M_{0}<K$
a. Show by substitution that the solution of the initial value problem is
$$M(t)=K\left(\frac{M_{0}}{K}\right)^{\operatorname{crp}(-n)}$$
b. Graph the solution for $M_{0}=100$ and $r=0.05$
c. Using the graph in part (b), estimate $\lim _{t \rightarrow \infty} M(t),$ the limiting size of the tumor.

Vipender Yadav
Vipender Yadav
Numerade Educator