Differential Equations and Linear Algebra 4th

Stephen W. Goode, Scott A. Annin

Chapter 1

First-Order Differential Equations

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Section 1

Differential Equations Everywhere

Problem 1

Determine the order of the differential equation.
$$\frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=e^{x}$$

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Problem 2

Determine the order of the differential equation.
$$\left(\frac{d y}{d x}\right)^{3}+y^{2}=\sin x$$

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Problem 3

Determine the order of the differential equation.
$$y^{\prime \prime}+x y^{\prime}+e^{x} y=y^{\prime \prime \prime}$$

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Problem 4

Determine the order of the differential equation.
$$\sin \left(y^{\prime \prime}\right)+x^{2} y^{\prime}+x y=\ln x$$

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Problem 5

Verify that, for $t>0, y(t)=\ln t$ is a solution to the differential equation
$$
2\left(\frac{d y}{d t}\right)^{3}=\frac{d^{3} y}{d t^{3}}
$$

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Problem 6

Verify that $y(x)=x /(x+1)$ is a solution to the differential equation
$$
y+\frac{d^{2} y}{d x^{2}}=\frac{d y}{d x}+\frac{x^{3}+2 x^{2}-3}{(1+x)^{3}}
$$

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Problem 7

Verify that $y(x)=e^{x} \sin x$ is a solution to the differential equation
$$
2 y \cot x-\frac{d^{2} y}{d x^{2}}=0
$$

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Problem 8

By writing Equation (1.1.7) in the form
$$
\frac{1}{T-T_{m}} \frac{d T}{d t}=-k
$$
and using $u^{-1} \frac{d u}{d t}=\frac{d}{d t}(\ln u),$ derive $(1.1 .8)$

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Problem 9

A glass of water whose temperature is $50^{\circ} \mathrm{F}$ is taken outside at noon on a day whose temperature is constant at $70^{\circ} \mathrm{F}$. If the water's temperature is $55^{\circ} \mathrm{F}$ at $2 \mathrm{p} . \mathrm{m} .,$ do you expect the water's temperature to reach $60^{\circ} \mathrm{F}$ before $4 \mathrm{p} . \mathrm{m}$. or after $4 \mathrm{p} . \mathrm{m} . ?$ Use Newton's law of cooling to explain your answer.

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Problem 10

On a cold winter day $\left(10^{\circ} \mathrm{F}\right),$ an object is brought outside from a $70^{\circ} \mathrm{F}$ room. If it takes 40 minutes for the object to cool from $70^{\circ} \mathrm{F}$ to $30^{\circ} \mathrm{F}$, did it take more or less than 20 minutes for the object to reach $50^{\circ} \mathrm{F}$ ? Use Newton's law of cooling to explain your answer.

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Problem 11

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$x^{2}+9 y^{2}=c$$

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Problem 12

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$y=c x^{2}$$

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Problem 13

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$y=c / x$$

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Problem 14

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$y=c x^{5}$$

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Problem 15

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$y=c e^{x}$$

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Problem 16

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.
$$y^{2}=2 x+c$$

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Problem 17

$m$ denotes a fixed nonzero constant, and $c$ is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.
$$y=c x^{m}$$

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Problem 18

$m$ denotes a fixed nonzero constant, and $c$ is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.
$$y=m x+c$$

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Problem 19

$m$ denotes a fixed nonzero constant, and $c$ is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.
$$y^{2}=m x+c$$

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Problem 20

$m$ denotes a fixed nonzero constant, and $c$ is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.
$$y^{2}+m x^{2}=c$$

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Problem 21

Consider the family of circles $x^{2}+y^{2}=2 c x .$ Show that the differential equation for determining the family of orthogonal trajectories is
$$
\frac{d y}{d x}=\frac{2 x y}{x^{2}-y^{2}}
$$

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Problem 22

We call a coordinate system $(u, v)$ orthogonal if its coordinate curves (the two families of curves $u=$ constant and $v=$ constant ) are orthogonal trajectories (for example, a Cartesian coordinate system or a polar coordinate system). Let $(u, v)$ be orthogonal coordinates, where $u=x^{2}+2 y^{2},$ and $x$ and $y$ are Cartesian coordinates. Find the Cartesian equation of the $v$ -coordinate curves, and sketch the $(u, v)$ coordinate system.

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Problem 23

Any curve with the property that whenever it intersects a curve of a given family it does so at an angle $a \neq \pi / 2$ is called an oblique trajectory of the given family. (See Figure 1.1.7.) Let $m_{1}$ (equal to tan $a_{1}$ ) denote the slope of the required family at the point $(x, y),$ and let $m_{2}$ (equal to tan $a_{2}$ ) denote the slope of the given family. Show that
$$
m_{1}=\frac{m_{2}-\tan a}{1+m_{2} \tan a}
$$
[Hint: From Figure $\left.1.1 .7, \tan a_{1}=\tan \left(a_{2}-a\right)\right] .$ Thus,
the equation of the family of oblique trajectories is obtained by solving
$$
\frac{d y}{d x}=\frac{m_{2}-\tan a}{1+m_{2} \tan a}
$$
(FIGURE CAN'T COPY)

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Problem 24

An object is released from rest at a height of 100 meters above the ground. Neglecting frictional forces, the subsequent motion is governed by the initial-value problem
$$
\frac{d^{2} y}{d t^{2}}=g, \quad y(0)=0, \quad \frac{d y}{d t}(0)=0
$$where $y(t)$ denotes the displacement of the object from its initial position at time $t$. Solve this initial-value problem and use your solution to determine the time when the object hits the ground.

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Problem 25

A five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, the subsequent motion is governed by the differential equation
$$
\frac{d^{2} y}{d t^{2}}=g
$$
If the object hits the ground 8 seconds after the boy releases it, find
(a) the time when the tennis ball reaches its maximum height.
(b) the maximum height of the tennis ball.

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Problem 26

A pyrotechnic rocket is to be launched vertically upwards from the ground. For optimal viewing, the rocket should reach a maximum height of 90 meters above the ground. Ignore frictional forces.
(a) How fast must the rocket be launched in order to achieve optimal viewing?
(b) Assuming the rocket is launched with the speed determined in part (a), how long after the rocket is launched will it reach its maximum height?

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Problem 27

Repeat Problem 26 under the assumption that the rocket is launched from a platform five meters above the ground.

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Problem 28

An object that is initially thrown vertically upward with a speed of 2 meters/second from a height of $h$ meters takes 10 seconds to reach the ground. Set up and solve the initial-value problem that governs the motion of the object, and determine $h$

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Problem 29

An object that is released from a height $h$ meters above the ground with a vertical velocity of $v_{0}$ meters/second hits the ground after $t_{0}$ seconds. Neglecting frictional forces, set up and solve the initial-value problem governing the motion, and use your solution to show that
$$
v_{0}=\frac{1}{2 t_{0}}\left(2 h-g t_{0}^{2}\right)
$$

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Problem 30

Verify that $y(t)=A \cos (\omega t-\phi)$ is a solution to the differential equation $(1.1 .21),$ where $A$ and $\omega$ are nonzero constants. Determine the constants $A$ and $\phi$ (with $|\phi|<\pi$ radians) in the particular case when the initial conditions are
$$
y(0)=a, \quad \frac{d y}{d t}(0)=0
$$

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Problem 31

Verify that
$$
y(t)=c_{1} \cos \omega t+c_{2} \sin \omega t
$$
is a solution to the differential equation (1.1.21). Show that the amplitude of the motion is
$$
A=\sqrt{c_{1}^{2}+c_{2}^{2}}
$$

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Problem 32

A heron has a birth mass of $3 \mathrm{g},$ and when fully grown its mass is 2700 g. Using equation $(1.1 .26)$ with $a=1.5$ determine the mass of the heron after
30 days.

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Problem 33

A rat has a birth mass of $8 \mathrm{g},$ and when fully grown its mass is 280 g. Using equation $(1.1 .26)$ with $a=0.25$ determine how many days it will take for the rat to reach $75 \%$ of its fully grown size.

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