Content check — Calculus Vol. 2, Chapter 4: Introduction to Differential Equations

Checkpoint 4.2 §4.1 source ↗

What is the order of the following differential equation?

(x^{4} - 3 x) y^{(5)} - (3 x^{2} + 1) y^{'} + 3 y = sin x cos x

Answer

5

Checkpoint 4.3 §4.1 source ↗

Find the particular solution to the differential equation

y^{'} = 4 x + 3

passing through the point $(1, 7),$ given that $y = 2 x^{2} + 3 x + C$ is a general solution to the differential equation.

Answer

y = 2 x^{2} + 3 x + 2

Checkpoint 4.5 §4.1 source ↗

Solve the initial-value problem

y^{'} = x^{2} - 4 x + 3 - 6 e^{x}, y (0) = 8 .

Answer

y =

\frac{1}{3} x^{3} -

2 x^{2} +

3 x -

6 e^{x} +

14

Checkpoint 4.6 §4.1 source ↗

Suppose a rock falls from rest from a height of

100

meters and the only force acting on it is gravity. Find an equation for the velocity

v (t)

as a function of time, measured in meters per second.

Answer

v (t) = −9.8 t

4-1 §4.1 source ↗

Determine the order of the following differential equations.

$y^{'} + y = 3 y^{2}$

Answer

1

4-3 §4.1 source ↗

Determine the order of the following differential equations.

$y ‴ + y^{″} y^{'} = 3 x^{2}$

Answer

3

4-5 §4.1 source ↗

Determine the order of the following differential equations.

$\frac{d y}{d t} = t$

Answer

1

4-7 §4.1 source ↗

Determine the order of the following differential equations.

${(\frac{d y}{d t})}^{2} +$ $8 \frac{d y}{d t} +$ $3 y =$ $4 t$

Answer

1

4-19 §4.1 source ↗

Verify the following general solutions and find the particular solution.

Find the particular solution to the differential equation $y^{'} = 3 x^{3}$ that passes through $(1, 4.75),$ given that $y = C + \frac{3 x^{4}}{4}$ is a general solution.

Answer

y = 4 + \frac{3 x^{4}}{4}

4-21 §4.1 source ↗

Verify the following general solutions and find the particular solution.

Find the particular solution to the differential equation $y^{'} = 2 x y$ that passes through $(0, \frac{1}{2}),$ given that $y = C e^{x^{2}}$ is a general solution.

Answer

y = \frac{1}{2} e^{x^{2}}

4-23 §4.1 source ↗

Verify the following general solutions and find the particular solution.

Find the particular solution to the differential equation $y^{'} x^{2} = y$ that passes through $(1, \frac{2}{e}),$ given that $y = C e^{− 1 / x}$ is a general solution.

Answer

y = 2 e^{− 1 / x}

4-25 §4.1 source ↗

Verify the following general solutions and find the particular solution.

Find the particular solution to the differential equation $\frac{d u}{d t} = tan u$ that passes through $(1, \frac{π}{2}),$ given that $u =$ ${sin}^{−1} (e^{C + t})$ is a general solution.

Answer

u =

{sin}^{−1} (e^{−1 + t})

4-27 §4.1 source ↗

Verify the following general solutions and find the particular solution.

Find the particular solution to the differential equation $y^{'} (1 - x^{2}) =$ $1 +$ $y$ that passes through $(0, −2),$ given that $y = C \frac{\sqrt{x + 1}}{\sqrt{1 - x}} - 1$ is a general solution.

Answer

y = - \frac{\sqrt{x + 1}}{\sqrt{1 - x}} - 1

4-29 §4.1 source ↗

For the following problems, find the general solution to the differential equation.

$y^{'} = ln x + tan x$

Answer

y =

C -

x +

x ln x -

ln (cos x)

4-31 §4.1 source ↗

For the following problems, find the general solution to the differential equation.

$y^{'} = 4^{x}$

Answer

y = C + \frac{4^{x}}{ln (4)}

4-33 §4.1 source ↗

For the following problems, find the general solution to the differential equation.

$y^{'} = 2 t \sqrt{t^{2} + 16}$

Answer

y =

\frac{2}{3} \sqrt{t^{2} + 16} (t^{2} + 16) +

C

4-35 §4.1 source ↗

For the following problems, find the general solution to the differential equation.

$x^{'} = t \sqrt{4 + t}$

Answer

x =

\frac{2}{15} \sqrt{4 + t} (3 t^{2} + 4 t - 32) +

C

4-37 §4.1 source ↗

For the following problems, find the general solution to the differential equation.

$y^{'} = \frac{y}{x}$

Answer

y = C x

4-39 §4.1 source ↗

Solve the following initial-value problems starting from $y (0) = 1$ and $y (0) = −1 .$ Draw both solutions on the same graph.

$\frac{d y}{d t} = − t$

Answer

y =

1 -

\frac{t^{2}}{2},

y =

-

\frac{t^{2}}{2} -

1

4-41 §4.1 source ↗

Solve the following initial-value problems starting from $y (0) = 1$ and $y (0) = −1 .$ Draw both solutions on the same graph.

$\frac{d y}{d t} = − y$

Answer

y =

e^{− t},

y =

− e^{− t}

4-43 §4.1 source ↗

Solve the following initial-value problems starting from $y_{0} = 10 .$ At what time does $y$ increase to $100$ or drop to $1 ?$

$\frac{d y}{d t} = 4 t$

Answer

y =

2 (t^{2} + 5),

t =

3 \sqrt{5}

4-45 §4.1 source ↗

Solve the following initial-value problems starting from $y_{0} = 10 .$ At what time does $y$ increase to $100$ or drop to $1 ?$

$\frac{d y}{d t} = −2 y$

Answer

y =

10 e^{−2 t},

t =

-

\frac{1}{2} ln (\frac{1}{10})

4-47 §4.1 source ↗

Solve the following initial-value problems starting from $y_{0} = 10 .$ At what time does $y$ increase to $100$ or drop to $1 ?$

$\frac{d y}{d t} = e^{−4 t}$

Answer

y =

\frac{1}{4} (41 - e^{−4 t}),

never

4-49 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

[T] $x y^{'} = y$

Answer

Solution changes from increasing to decreasing at

y (0) = 0

4-51 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

[T] $y^{'} = x + y$ (Hint: $y = C e^{x} - x - 1$ is the general solution)

Answer

Solution changes from increasing to decreasing at

y (0) = 0

4-53 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

Find the general solution to describe the velocity of a ball of mass $1 lb$ that is thrown upward at a rate $a$ ft/sec.

Answer

v (t) = −32 t + a

4-55 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

You throw two objects with differing masses $m_{1}$ and $m_{2}$ upward into the air with the same initial velocity $a$ ft/s. What is the difference in their velocity after $1$ second?

Answer

0

ft/s

4-57 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

[T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars, where $g = - 9.8 m / s^{2}$ .

Answer

52.354

meters

4-59 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

[T] For the car in the preceding problem, find the expression for the distance the car has traveled in time $t,$ assuming an initial distance of $0 .$ How long does it take the car to travel $100$ miles? Round your answer to hours and minutes.

Answer

x =

50 t -

\frac{15}{π^{2}} cos (π t) +

\frac{3}{π^{2}},

2

hours

1

minute

4-61 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

Substitute $y = B e^{3 t}$ into $y^{'} - y = 8 e^{3 t}$ to find a particular solution.

Answer

y = 4 e^{3 t}

4-63 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

Substitute $y = a + b t + c t^{2}$ into $y^{'} + y = 1 + t^{2}$ to find a particular solution.

Answer

y = 3 - 2 t + t^{2}

4-65 §4.1 source ↗

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from $y (t = 0) = −10$ to $y (t = 0) = 10$ increasing by $2 .$ Is there some critical point where the behavior of the solution begins to change?

Solve $y^{'} = e^{k t}$ with the initial condition $y (0) = 0$ and solve $y^{'} = 1$ with the same initial condition. As $k$ approaches $0,$ what do you notice?

Answer

y = \frac{1}{k} (e^{k t} - 1)

and

y = x

Checkpoint 4.7 §4.2 source ↗

Create a direction field for the differential equation

y' = x^{2} - y^{2}

and sketch a solution curve passing through the point

(−1, 2) .

Answer

A graph of the direction field for the differential equation y’ = x ^ 2 – y ^ 2. Along y = x and y = -x, the lines are horizontal. On either side of y = x and y = -x, the lines slant and direct solutions along those two functions. The rest of the lines are vertical. The solution going through (-1, 2) is shown. It curves down from about (-2.75, 10), through (-1, 2) and about (0, 1.5), and then up along the diagonal to (10, 10).

Checkpoint 4.8 §4.2 source ↗

Create a direction field for the differential equation

y' =

(x + 5) (y + 2) (y^{2} - 4 y + 4)

and identify any equilibrium solutions. Classify each of the equilibrium solutions as stable, unstable, or semi-stable.

Answer

A direction field with arrows pointing to the right at y = -4 and y = 4. The arrows point up for y greater than -4 and down for y less than -4. Close to y = 4, the arrows are more horizontal, but the further away, the more vertical they become.

The equilibrium solutions are $y = −2$ and $y = 2 .$ For this equation, $y = −2$ is an unstable equilibrium solution, and $y = 2$ is a semi-stable equilibrium solution.

Checkpoint 4.9 §4.2 source ↗

Consider the initial-value problem

y^{'} = x^{3} + y^{2}, y (1) = −2 .

Using a step size of $0.1,$ generate a table with approximate values for the solution to the initial-value problem for values of $x$ between $1$ and $2 .$

Answer

$n$	$x_{n}$	$y_{n} =$ $y_{n - 1} +$ $h f (x_{n - 1}, y_{n - 1})$
$0$	$1$	$−2$
$1$	$1.1$	$y_{1} =$ $y_{0} +$ $h f (x_{0}, y_{0}) =$ $−1.5$
$2$	$1.2$	$y_{2} =$ $y_{1} +$ $h f (x_{1}, y_{1}) =$ $−1.1419$
$3$	$1.3$	$y_{3} =$ $y_{2} +$ $h f (x_{2}, y_{2}) =$ $−0.8387$
$4$	$1.4$	$y_{4} =$ $y_{3} +$ $h f (x_{3}, y_{3}) =$ $−0.5487$
$5$	$1.5$	$y_{5} =$ $y_{4} +$ $h f (x_{4}, y_{4}) =$ $−0.2442$
$6$	$1.6$	$y_{6} =$ $y_{5} +$ $h f (x_{5}, y_{5}) =$ $0.0993$
$7$	$1.7$	$y_{7} =$ $y_{6} +$ $h f (x_{6}, y_{6}) =$ $0.5099$
$8$	$1.8$	$y_{8} =$ $y_{7} +$ $h f (x_{7}, y_{7}) =$ $1.0272$
$9$	$1.9$	$y_{9} =$ $y_{8} +$ $h f (x_{8}, y_{8}) =$ $1.7159$
$10$	$2$	$y_{10} =$ $y_{9} +$ $h f (x_{9}, y_{9}) =$ $2.6962$

4-67 §4.2 source ↗

For the following problems, use the direction field below from the differential equation $y' = −2 y .$ Sketch the graph of the solution for the given initial conditions.

A direction field with horizontal arrows pointing to the right at 0. The arrows above the x axis point down and to the right. The further away from the x axis, the steeper the arrows are, and the closer to the x axis, the flatter the arrows are. Likewise, the arrows below the x axis point up and to the right. The further away from the x axis, the steeper the arrows are, and the closer to the x axis, the flatter the arrows are.

$y (0) = 0$

Answer

A graph of the given direction field with a flat line drawn on the axis. The arrows point up for y less than 0 and down for y greater than 0. The closer they are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they become.

4-69 §4.2 source ↗

For the following problems, use the direction field below from the differential equation $y' = −2 y .$ Sketch the graph of the solution for the given initial conditions.

Are there any equilibria? What are their stabilities?

Answer

y = 0

is a stable equilibrium

4-71 §4.2 source ↗

For the following problems, use the direction field below from the differential equation $y' = y^{2} - 2 y .$ Sketch the graph of the solution for the given initial conditions.

A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y greater than 2 and for y less than 0. The arrows point down for 0 less than y less than 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are.

$y (0) = 1$

Answer

4-73 §4.2 source ↗

For the following problems, use the direction field below from the differential equation $y' = y^{2} - 2 y .$ Sketch the graph of the solution for the given initial conditions.

Are there any equilibria? What are their stabilities?

Answer

y = 0

is a stable equilibrium and

y = 2

is unstable

4-75 §4.2 source ↗

Draw the direction field for the following differential equations, then solve the differential equation.

$y' = e^{t}$

Answer

General solution is $y = e^{t} + C$ .

A direction field over the four quadrants. As t goes from 0 to infinity, the arrows become more and more vertical after being horizontal closer to x = 0.

4-77 §4.2 source ↗

Draw the direction field for the following differential equations, then solve the differential equation.

$\frac{d y}{d t} = t e^{t}$

Answer

General solution is $y = e^{t} (t - 1) + C$ .

A direction field over [-2, 2] in the x and y axes. The arrows point slightly down and to the right over [-2, 0] and gradually become vertical over [0, 2].

4-79 §4.2 source ↗

Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?

$y' = y^{2} - 1$

Answer

A direction field with horizontal arrows pointing to the right at y = 1 and y = -1. The arrows point up for y less than -1 and y greater than 1. The arrows point down for -1 less than y less than 1. The closer the arrows are to these lines, the more horizontal they are, and the further away they are, the more vertical they are.

4-81 §4.2 source ↗

Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?

$y' = 1 - y^{2} - x^{2}$

Answer

A direction field with arrows pointing down and to the right for nearly all points in [-2, 2] on the x and y axes. Close to the origin, the arrows become more horizontal, point to the upper right, become more horizontal, and then point down to the right again.

4-83 §4.2 source ↗

Draw the directional field for the following differential equations. What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have?

$y' = 3 y + x y$

Answer

A direction field with horizontal arrows pointing to the right on the x axis and x = -3. Above the x axis and for x less than -3, the arrows point down. For x greater than -3, the arrows point up. Below the x axis and for x less than -3, the arrows point up. For x greater than -3, the arrows point down. The further away from the x axis and x = -3, the arrows become more vertical, and the closer they become, the more horizontal they become.

4-85 §4.2 source ↗

Match the direction field with the given differential equations. Explain your selections.

A direction field with arrows pointing down and to the right in quadrants two and three. After crossing the y axis, the arrows change direction and point up to the right.

A direction field with horizontal arrows pointing to the left in quadrants two and three. In crossing the y axis, the arrows switch and point upward in quadrants one and four.

A direction field with horizontal arrows pointing to the right on the x axis. Above, the arrows point down and to the right, and below, the arrows point up and to the right. The further from the x axis, the more vertical the arrows become.

A direction field with horizontal arrows on the x and y axes. The arrows point down and to the right in quadrants one and three. They point up and to the right in quadrants two and four.

A direction field with arrows pointing up in quadrants two and three, to the right on the y axis, and down in quadrants one and four.

$y' = −3 t$

Answer

E

4-87 §4.2 source ↗

Match the direction field with the given differential equations. Explain your selections.

$y' = \frac{1}{2} y + t$

Answer

A

4-89 §4.2 source ↗

Match the direction field with the given differential equations. Explain your selections.

A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up, and in quadrants two and four, they point down.

A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants one and three, the arrows point up and to the right, and in quadrants two and four, the arrows point down and to the right.

A direction field with horizontal arrows pointing to the right on the x and y axes. In quadrants two and three, the arrows point down, and in quadrants one and four, the arrows point up.

A direction field with horizontal arrows pointing to the right on the x axis. The arrows point up and to the right in all quadrants. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they are.

A direction field with horizontal arrows on the y axis. The arrows are also more horizontal closer to y = 1.5, y = -1.5, and the y axis. For y greater than 1.5 and x less than 0, for y less than -1.5 and x less than 0, and for -1.5 less than y less than 1.5 and x greater than 0-, the arrows point down. For y greater than 1.5 and x greater than 0, for y less than -1.5, for y less than -1.5 and x greater than 0, and for -1.5 less than y less than 1.5 and x less than 0, the arrows point up.

$y' = t sin y$

Answer

B

4-91 §4.2 source ↗

Match the direction field with the given differential equations. Explain your selections.

$y' = t tan y$

Answer

A

4-93 §4.2 source ↗

Match the direction field with the given differential equations. Explain your selections.

$y' = y^{2} t^{3}$

Answer

C

4-95 §4.2 source ↗

Estimate the following solutions using Euler’s method with $n = 5$ steps over the interval $t = [0, 1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

$y' =$ $t^{2},$ $y (0) =$ $2$

Answer

2.24,

exact:

2

4-97 §4.2 source ↗

Estimate the following solutions using Euler’s method with $n = 5$ steps over the interval $t = [0, 1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

$y^{'} =$ $y +$ $t^{2},$ $y (0) =$ $3 .$ Exact solution is $y =$ $5 e^{t} -$ $2 -$ $t^{2} -$ $2 t$

Answer

7.739264,

exact:

5 (e - 1)

4-99 §4.2 source ↗

Estimate the following solutions using Euler’s method with $n = 5$ steps over the interval $t = [0, 1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

[T] $y' =$ $e^{(x + y)},$ $y (0) =$ $−1 .$ Exact solution is $y =$ $− ln (e + 1 - e^{x})$

Answer

−0.2535

exact:

0

4-101 §4.2 source ↗

Estimate the following solutions using Euler’s method with $n = 5$ steps over the interval $t = [0, 1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

$y^{'} =$ $2^{x},$ $y (0) =$ $0,$ Exact solution is $y = \frac{2^{x} - 1}{ln (2)}$

Answer

1.345,

exact:

\frac{1}{ln (2)}

4-103 §4.2 source ↗

Estimate the following solutions using Euler’s method with $n = 5$ steps over the interval $t = [0, 1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?

$y^{'} =$ $−5 t,$ $y (0) =$ $−2 .$ Exact solution is $y = - \frac{5}{2} t^{2} - 2$

Answer

−4,

exact:

− 1 / 2

4-105 §4.2 source ↗

Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. $S$ represents the size of the susceptible population, and $I$ represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability $c$ that the susceptible person will become infected. Each infected person recovers from the infection at a rate $r$ and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, $N .$ The differential equations that model these population sizes are

\begin{matrix} S' = r I - c S I and \\ I' = c S I - r I . \end{matrix}

Here $c$ represents the contact rate and $r$ is the recovery rate.

Assuming the parameters are $c = 0.5, N = 5,$ and $r = 0.5,$ draw the resulting directional field.

Answer

A direction field with horizontal arrows pointing to the right on the x axis and at y = 4. The arrows below the x axis and above y = 4 point down and to the right. The arrows between the x axis and y = 4 point up and to the right.

4-107 §4.2 source ↗

Refer to your answer for Exercise 4.106 to solve the following exercises.

Solve the initial-value problem for the exact solution.

Answer

y' = 2 e^{t^{2} / 2}

4-109 §4.2 source ↗

Refer to your answer for Exercise 4.106 to solve the following exercises.

Compute using $h = 1$

Answer

2

4-111 §4.2 source ↗

Refer to your answer for Exercise 4.106 to solve the following exercises.

Compute using [T] $h = 0.001$

Answer

3.2756

4-113 §4.2 source ↗

Refer to your answer for Exercise 4.106 to solve the following exercises.

[T] Evaluate the exact solution at $t = 1 .$ Make a table of errors for the relative error between the Euler’s method solution and the exact solution. How much does the error change? Can you explain?

Answer

$2 \sqrt{e}$

Step Size	Relative Error
$h = 0.1$	$0.3935$
$h = 0.01$	$0.06163$
$h = 0.001$	$0.006612$
$h = 0.0001$	$0.0006661$

4-115 §4.2 source ↗

Consider the initial-value problem $y' =$ $−2 y,$ $y (0) =$ $2 .$

Draw the directional field of this differential equation.

Answer

A direction field with horizontal arrows pointing to the right on the x axis. Above the x axis, the arrows point down and to the right. Below the x axis, the arrows point up and to the right. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are from the x axis, the more vertical the arrows are.

4-117 §4.2 source ↗

Consider the initial-value problem $y' =$ $−2 y,$ $y (0) =$ $2 .$

[T] By calculator or computer, approximate the solution using Euler’s Method at $t = 10$ using $h = 100 .$

Answer

4.0741 e^{−10}

Checkpoint 4.10 §4.3 source ↗

Use the method of separation of variables to find a general solution to the differential equation

y' =

2 x y +

3 y -

4 x -

6 .

Answer

y = 2 + C e^{x^{2} + 3 x}

Checkpoint 4.11 §4.3 source ↗

Find the solution to the initial-value problem

6 y' = (2 x + 1) (y^{2} - 2 y - 8), y (0) = −3

using the method of separation of variables.

Answer

y =

\frac{4 + 14 e^{x^{2} + x}}{1 - 7 e^{x^{2} + x}}

Checkpoint 4.12 §4.3 source ↗

A tank contains

3

kilograms of salt dissolved in

75

liters of water. A salt solution of

0.4 kg salt/L

is pumped into the tank at a rate of

6 L/min

and is drained at the same rate. Solve for the salt concentration at time

t .

Assume the tank is well mixed at all times.

Answer

Initial value problem:

$\frac{d u}{d t} =$ $2.4 -$ $\frac{2 u}{25},$ $u (0) =$ $3$

$Solution: u (t) =$ $30 -$ $27 e^{− 2 t / 25}$
$Concentration: 30 - 27 e^{\frac{- 2 t}{25}}$

Checkpoint 4.13 §4.3 source ↗

A cake is removed from the oven after baking thoroughly, and the temperature of the cake when it comes out of the oven is $450 ° F .$ The temperature of the kitchen is $70 ° F,$ and after $10$ minutes the temperature of the cake is $330 ° F .$

a. Write the appropriate initial-value problem to describe this situation.

b. Solve the initial-value problem for $T (t) .$

c. How long will it take until the temperature of the cake is within $5 ° F$ of room temperature?

Answer

a. Initial value problem
$\frac{d T}{d t} =$ $k (T - 70),$ $T (0) =$ $450$

b. $T (t) =$ $70 +$ $380 e^{k t}$

c. Approximately $114$ minutes.

4-119 §4.3 source ↗

Solve the following initial-value problems with the initial condition $y_{0} = 0$ and graph the solution.

$\frac{d y}{d t} = y + 1$

Answer

y = e^{t} - 1

4-121 §4.3 source ↗

Solve the following initial-value problems with the initial condition $y_{0} = 0$ and graph the solution.

$\frac{d y}{d t} = –y + 1$

Answer

y = 1 + C e^{− t}

4-123 §4.3 source ↗

Find the general solution to the differential equation.

$x^{2} y' = (x + 1) y$

Answer

y = C x e^{−1 / x}

4-125 §4.3 source ↗

Find the general solution to the differential equation.

$y' = 2 x y^{2}$

Answer

y = \frac{1}{C - x^{2}}

4-127 §4.3 source ↗

Find the general solution to the differential equation.

$2 x \frac{d y}{d x} = y^{2}$

Answer

y = - \frac{2}{C + ln x}

4-129 §4.3 source ↗

Find the general solution to the differential equation.

$(1 + x) y' =$ $(x + 2) (y - 1)$

Answer

y =

C e^{x} (x + 1) +

1

4-131 §4.3 source ↗

Find the general solution to the differential equation.

$t \frac{d y}{d t} = \sqrt{1 - y^{2}}$

Answer

y =

sin (ln t + C)

4-133 §4.3 source ↗

Find the solution to the initial-value problem.

$y' =$ $e^{y - x},$ $y (0) =$ $0$

Answer

y = − ln (e^{− x})

4-135 §4.3 source ↗

Find the solution to the initial-value problem.

$\frac{d y}{d x} =$ $y^{3} x e^{x^{2}},$ $y (0) =$ $1$

Answer

y = \frac{1}{\sqrt{2 - e^{x^{2}}}}

4-137 §4.3 source ↗

Find the solution to the initial-value problem.

$y' =$ $\frac{x}{{sech}^{2} y},$ $y (0) =$ $0$

Answer

y =

{tanh}^{−1} (\frac{x^{2}}{2})

4-139 §4.3 source ↗

Find the solution to the initial-value problem.

$\frac{d x}{d t} =$ $ln (t) \sqrt{1 - x^{2}},$ $x (1) =$ $0$

Answer

x =

sin (1 - t + t ln t)

4-141 §4.3 source ↗

Find the solution to the initial-value problem.

$y' =$ $e^{y} 5^{x},$ $y (0) =$ $ln (ln (5))$

Answer

y =

ln (ln (5)) -

ln (2 - 5^{x})

4-143 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y' = 1 - 2 y$

Answer

$y = C e^{−2 x} + \frac{1}{2}$

A direction field with horizontal arrows pointing to the right at y = 0.5. Above 0.5, the arrows slope down and to the right and are increasingly vertical the further they are from y = 0.5 Below0.5, the arrows slope up and to the right and are increasingly vertical the further they are from y = 0.5.

4-145 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y' = y^{3} e^{x}$

Answer

$y = \frac{1}{\sqrt{2} \sqrt{C - e^{x}}}$

A direction field with arrows pointing to the right. They are horizontal at the y axis. The further the arrows are from the axis, the more vertical they become. They point up above the x axis and down below the x axis.

4-147 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] $y' = y ln (x)$

Answer

$y = C e^{− x} x^{x}$

A direction field with arrows pointing to the right. The arrows are flat on y = 1. The further the arrows are from that, the steeper they become. They point up above that line and down below that line.

4-149 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

A drug is administered intravenously to a patient at a rate $r$ mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, $d$ . Set up and solve the differential equation, assuming there is no drug initially present in the body.

Answer

y =

\frac{r}{d} (1 - e^{− d t})

4-151 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

A tank contains $1$ kilogram of salt dissolved in $100$ liters of water. A salt solution of $0.1$ kg salt/L is pumped into the tank at a rate of $2$ L/min and is drained at the same rate. Solve for the salt concentration at time $t .$ Assume the tank is well mixed.

Answer

y (t) =

10 -

9 e^{− x / 50}

4-153 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

[T] For the preceding problem, find how much salt is in the tank $1$ hour after the process begins.

Answer

134.3

kilograms

4-155 §4.3 source ↗

For the following problems, use a software program or your calculator to generate the directional fields. Solve explicitly and draw solution curves for several initial conditions. Are there some critical initial conditions that change the behavior of the solution?

For the preceding problem, determine how long it takes the tank to drain.

Answer

720

seconds

4-157 §4.3 source ↗

For the following problems, use Newton’s law of cooling.

[T] The liquid base of an ice cream has an initial temperature of $210 ° F$ before it is placed in a freezer with a constant temperature of $20 ° F .$ After $2$ hours, the temperature of the ice-cream base has decreased to $170 ° F .$ At what time will the ice cream be ready to eat? (Assume $30 ° F$ is the optimal eating temperature.)

Answer

24

hours

57

minutes

4-159 §4.3 source ↗

For Exercises 159—162, assume a cooling constant of $k = - 0.125$ and assume time $t$ is in minutes.

You have a cup of coffee at temperature $70 ° C$ and the ambient temperature in the room is $20 ° C .$ Assuming a cooling rate $k of 0.125,$ write and solve the differential equation to describe the temperature of the coffee with respect to time.

Answer

T (t) =

20 +

50 e^{−0.125 t}

4-161 §4.3 source ↗

For Exercises 159—162, assume a cooling constant of $k = - 0.125$ and assume time $t$ is in minutes.

You have a cup of coffee at temperature $70 ° C$ and you immediately pour in $1$ part milk to $5$ parts coffee. The milk is initially at temperature $1 ° C .$ Write and solve the differential equation that governs the temperature of this coffee.

Answer

T (t) =

20 +

38.5 e^{−0.125 t}

4-163 §4.3 source ↗

For Exercises 159—162, assume a cooling constant of $k = - 0.125$ and assume time $t$ is in minutes.

Solve the generic problem $y' = a y + b$ with initial condition $y (0) = c .$

Answer

y =

(c + \frac{b}{a}) e^{a x} -

\frac{b}{a}

4-165 §4.3 source ↗

For Exercises 159—162, assume a cooling constant of $k = - 0.125$ and assume time $t$ is in minutes.

Assume an initial nutrient amount of $I$ kilograms in a tank with $L$ liters. Assume a concentration of $c$ kg/L being pumped in at a rate of $r$ L/min. The tank is well mixed and is drained at a rate of $r$ L/min. Find the equation describing the amount of nutrient in the tank.

Answer

y (t) =

c L +

(I - c L) e^{− r t / L}

4-167 §4.3 source ↗

For Exercises 159—162, assume a cooling constant of $k = - 0.125$ and assume time $t$ is in minutes.

Leaves accumulate on the forest floor at a rate of $4$ g/cm²/yr. These leaves decompose at a rate of $10 %$ per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?

Answer

y =

40 (1 - e^{−0.1 t}),

40

g/cm²

Checkpoint 4.14 §4.4 source ↗

A population of rabbits in a meadow is observed to be $200$ rabbits at time $t = 0 .$ After a month, the rabbit population is observed to have increased by $4 % .$ Using an initial population of $200$ and a growth rate of $0.04,$ with a carrying capacity of $750$ rabbits,

a. Write the logistic differential equation and initial condition for this model.

b. Draw a slope field for this logistic differential equation, and sketch the solution corresponding to an initial population of $200$ rabbits.

c. Solve the initial-value problem for $P (t) .$

d. Use the solution to predict the population after $1$ year.

Answer

a.

$\frac{d P}{d t} =$ $0.04 (1 - \frac{P}{750}),$ $P (0) =$ $200$

b.

A direction field with horizontal lines on the x axis and y = 15. The other lines are vertical, except for those curving into the x axis and y = 15. A solution is drawn that crosses the y axis at about (0, 4) and asymptotically approaches y = 15.

c.

$P (t) =$ $\frac{3000 e^{.04 t}}{11 + 4 e^{.04 t}}$

d.

After $12$ months, the population will be $P (12) \approx 278$ rabbits.

4-169 §4.4 source ↗

For the following problems, consider the logistic equation in the form $P' = C P - P^{2} .$ Draw the directional field and find the stability of the equilibria.

$C = 0$

Answer

A direction field with arrows pointing to the right. The arrows are horizontal along the x axis. The arrows point down above the x axis and below the x axis. The further away the arrows are from the x axis, the more vertical the lines become.

$P = 0$ semi-stable

4-171 §4.4 source ↗

For the following problems, consider the logistic equation in the form $P' = C P - P^{2} .$ Draw the directional field and find the stability of the equilibria.

Solve the logistic equation for $C = 10$ and an initial condition of $P (0) = 2 .$

Answer

P =

\frac{10 e^{10 x}}{e^{10 x} + 4}

4-173 §4.4 source ↗

For the following problems, consider the logistic equation in the form $P' = C P - P^{2} .$ Draw the directional field and find the stability of the equilibria.

A population of deer inside a park has a carrying capacity of $200$ and a growth rate of $2 % .$ If the initial population is $50$ deer, what is the population of deer at any given time?

Answer

P (t) =

\frac{10000 e^{0.02 t}}{150 + 50 e^{0.02 t}}

4-175 §4.4 source ↗

For the following problems, consider the logistic equation in the form $P' = C P - P^{2} .$ Draw the directional field and find the stability of the equilibria.

[T] Bacteria grow at a rate of $20 %$ per hour in a petri dish. If there is initially one bacterium and a carrying capacity of $1$ million cells, how long does it take to reach $500,000$ cells?

Answer

69

hours

5

minutes

4-177 §4.4 source ↗

For the following problems, consider the logistic equation in the form $P' = C P - P^{2} .$ Draw the directional field and find the stability of the equilibria.

[T] Two monkeys are placed on an island. After $5$ years, there are $8$ monkeys, and the estimated carrying capacity is $25$ monkeys. When does the population of monkeys reach $16$ monkeys?

Answer

8

years

11

months

4-179 §4.4 source ↗

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

[T] The population of trout in a pond is given by $P' =$ $0.4 P (1 - \frac{P}{10000}) -$ $400,$ where $400$ trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?

Answer

A direction field with arrows down for P less than 1,000, pointing up for 1,000 less than P less than 8,500, and pointing down for P greater than 8,500. Right above P = 8,500, the arrows point down and to the right.

4-181 §4.4 source ↗

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

[T] For the preceding problem, use software to generate a directional field for the value $f = 400 .$ What are the stabilities of the equilibria?

Answer

A direction field with arrows pointing down and to the right. Around y = 4,000, the arrows are more horizontal. The further the arrows are from this line, the more vertical the arrows become.

$P_{1}$ semi-stable

4-183 §4.4 source ↗

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

[T] For the preceding problems, consider the case where a certain number of fish are added to the pond, or $f = −200 .$ What are the nonnegative equilibria and their stabilities?

Answer

A direction field with arrows pointing up for P less than 10,000 and arrows pointing down for P greater than 10,000.

$P_{2} > 0$ stable

4-185 §4.4 source ↗

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant $k,$ as

$P' =$ $0.4 P (1 - \frac{P}{10000}) -$ $k P .$

[T] Use software or a calculator to draw directional fields for $k = 0.4 .$ What are the nonnegative equilibria and their stabilities?

Answer

A direction field with arrows pointing to the right at P = 0. Below 0, the arrows point down and to the right. Above 0, the arrows point down and to the right. The further away from 0, the more vertical the arrows become.

$P_{1} = 0$ is semi-stable

4-187 §4.4 source ↗

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant $k,$ as

$P' =$ $0.4 P (1 - \frac{P}{10000}) -$ $k P .$

Solve this equation, assuming a value of $k = 0.05$ and an initial condition of $2000$ fish.

Answer

P t =

\frac{3500}{4 + 3 e^{- 035 t}}

4-189 §4.4 source ↗

The following problems add in a minimal threshold value for the species to survive, $T,$ which changes the differential equation to $P' (t) =$ $r P (1 - \frac{P}{K}) (1 - \frac{T}{P}) .$

Draw the directional field of the threshold logistic equation, assuming $K =$ $10,$ $r =$ $0.1,$ $T =$ $2 .$ When does the population survive? When does it go extinct?

Answer

A direction field with arrows pointing horizontally to the right along y = 2 and y = 10. For P less than 2, the arrows point down and to the right. For 2 less than P less than 10, the arrows point up and to the right. For P greater than 10, the arrows point down and to the right. The further the arrows are from 2 and 10, the steeper they become, and the closer they are from 2 and 10, the more horizontal the arrows become.

4-191 §4.4 source ↗

The following problems add in a minimal threshold value for the species to survive, $T,$ which changes the differential equation to $P' (t) =$ $r P (1 - \frac{P}{K}) (1 - \frac{T}{P}) .$

Bengal tigers in a conservation park have a carrying capacity of $100$ and need a minimum of $10$ to survive. If they grow in population at a rate of $1 %$ per year, with an initial population of $15$ tigers, solve for the number of tigers present.

Answer

P (t) =

\frac{850 + 500 e^{0.009 t}}{85 + 5 e^{0.009 t}}

4-193 §4.4 source ↗

The following problems add in a minimal threshold value for the species to survive, $T,$ which changes the differential equation to $P' (t) =$ $r P (1 - \frac{P}{K}) (1 - \frac{T}{P}) .$

The population of Florida panthers in the southwestern tip of Florida has an estimated carrying capacity of $250$ and grows at a rate of $0.25 %$ per year and there must be $25$ for the population to survive. With an initial population of $30$ Florida panthers, how many years will it take to get the Florida panthers off the endangered species list (at least $100) ?$

Answer

13

years months

4-195 §4.4 source ↗

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

Assume that for a population, $K = 1000$ and $α = 0.05 .$ Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?

Answer

A direction field with arrows pointing down and to the right for P less than 0, up for 0 less than P less than 1,000, and down for P greater than 1,000. The further the arrows are from P = 0 and P = 1,000, the more vertical they become, and the closer they are, the more horizontal they are.

4-197 §4.4 source ↗

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

[T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day $1$ and assuming $α = 0.1$ and a carrying capacity of $10$ million cells, how long does it take to reach “detection” stage at $5$ million cells?

Answer

31.465

days

4-199 §4.4 source ↗

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

[T] It is estimated that the world human population reached $3$ billion people in $1959$ and $6$ billion in $1999 .$ Assuming a carrying capacity of $16$ billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached $7$ billion. Was logistic growth or Gompertz growth more accurate, considering world population reached $7$ billion on October $31, 2011 ?$

Answer

September

2008

4-201 §4.4 source ↗

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

When does population increase the fastest in the threshold logistic equation $P' (t) =$ $r P (1 - \frac{P}{K}) (1 - \frac{T}{P}) ?$

Answer

\frac{K + T}{2}

4-203 §4.4 source ↗

Below is a table of the populations of whooping cranes in the wild from $1940 to 2000 .$ The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of $10,000$ cranes. Fit the data assuming years since $1940$ (so your initial population at time $0$ would be $22$ cranes).

Year (years since conservation began)	Whooping Crane Population
$1940 (0)$	$22$
$1950 (10)$	$31$
$1960 (20)$	$36$
$1970 (30)$	$57$
$1980 (40)$	$91$
$1990 (50)$	$159$
$2000 (60)$	$256$

Source: https://www.savingcranes.org/images/stories/site_images/conservation/whooping_crane/pdfs/historic_wc_numbers.pdf

Find the equation and parameter $r$ that best fit the data for the logistic equation.

Answer

r = 0.0405

4-205 §4.4 source ↗

Below is a table of the populations of whooping cranes in the wild from $1940 to 2000 .$ The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of $10,000$ cranes. Fit the data assuming years since $1940$ (so your initial population at time $0$ would be $22$ cranes).

Year (years since conservation began)	Whooping Crane Population
$1940 (0)$	$22$
$1950 (10)$	$31$
$1960 (20)$	$36$
$1970 (30)$	$57$
$1980 (40)$	$91$
$1990 (50)$	$159$
$2000 (60)$	$256$

Source: https://www.savingcranes.org/images/stories/site_images/conservation/whooping_crane/pdfs/historic_wc_numbers.pdf

Find the equation and parameter $α$ that best fit the data for the Gompertz equation.

Answer

α = 0.0081

4-207 §4.4 source ↗

Below is a table of the populations of whooping cranes in the wild from $1940 to 2000 .$ The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of $10,000$ cranes. Fit the data assuming years since $1940$ (so your initial population at time $0$ would be $22$ cranes).

Year (years since conservation began)	Whooping Crane Population
$1940 (0)$	$22$
$1950 (10)$	$31$
$1960 (20)$	$36$
$1970 (30)$	$57$
$1980 (40)$	$91$
$1990 (50)$	$159$
$2000 (60)$	$256$

Source: https://www.savingcranes.org/images/stories/site_images/conservation/whooping_crane/pdfs/historic_wc_numbers.pdf

Using the three equations found in the previous problems, estimate the population in $2010$ (year $70$ after conservation). The real population measured at that time was $437 .$ Which model is most accurate?

Answer

Logistic:

361,

Threshold:

436,

Gompertz:

309 .

Checkpoint 4.15 §4.5 source ↗

Put the equation

\frac{(x + 3) y'}{2 x - 3 y - 4} =

5

into standard form and identify

p (x)

and

q (x) .

Answer

y' +

\frac{15}{x + 3} y =

\frac{10 x - 20}{x + 3};

p (x) =

\frac{15}{x + 3}

and

q (x) =

\frac{10 x - 20}{x + 3}

Checkpoint 4.16 §4.5 source ↗

Find the general solution to the differential equation

(x - 2) y' +

y =

3 x^{2} +

2 x .

Assume

x > 2 .

Answer

y =

\frac{x^{3} + x^{2} + C}{x - 2}

Checkpoint 4.17 §4.5 source ↗

Solve the initial-value problem

y' -

2 y =

4 x +

3

y (0) =

−2 .

Answer

y =

- 2 x - \frac{5}{2} +

\frac{1}{2} e^{2 x}

Checkpoint 4.18 §4.5 source ↗

The weight of a penny is $2.5$ grams (United States Mint, “Coin Specifications,” accessed April 9, 2015, http://www.usmint.gov/about_the_mint/?action=coin_specifications), and the upper observation deck of the Empire State Building is $369$ meters above the street. Since the penny is a small and relatively smooth object, air resistance acting on the penny is actually quite small. We assume the air resistance is numerically equal to $0.0025 v .$ Furthermore, the penny is dropped with no initial velocity imparted to it.

a. Set up an initial-value problem that represents the falling penny.

b. Solve the problem for $v (t) .$

c. What is the terminal velocity of the penny (i.e., calculate the limit of the velocity as $t$ approaches infinity)?

Answer

a. $\begin{matrix} \frac{d v}{d t} & = & − v - 9.8 \\ v (0) & = & 0 \end{matrix}$

b. $v (t) =$ $9.8 (e^{− t} - 1)$

c. $lim_{t \to \infty} v (t) =$ $lim_{t \to \infty} (9.8 (e^{− t} - 1)) =$ $−9.8 m/s \approx$ $-$ $21.922 mph$

Checkpoint 4.19 §4.5 source ↗

A circuit has in series an electromotive force given by

E = 20 sin 5 t

V, a capacitor with capacitance

0.02 F,

and a resistor of

8 Ω .

If the initial charge is

4 C,

find the charge at time

t > 0 .

Answer

Initial-value problem:

$8 q^{'} +$ $\frac{1}{0.02} q =$ $20 sin 5 t,$ $q (0) =$ $4$

$q (t) =$ $\frac{10 sin 5 t - 8 cos 5 t + 172 e^{−6.25 t}}{41}$

4-209 §4.5 source ↗

Are the following differential equations linear? Explain your reasoning.

$\frac{d y}{d t} = t y$

Answer

Yes

4-211 §4.5 source ↗

Are the following differential equations linear? Explain your reasoning.

$y' = x^{3} + e^{x}$

Answer

Yes

4-213 §4.5 source ↗

Write the following first-order differential equations in standard form.

$y' = x^{3} y + sin x$

Answer

y' - x^{3} y = sin x

4-215 §4.5 source ↗

Write the following first-order differential equations in standard form.

$− x y' =$ $(3 x + 2) y +$ $x e^{x}$

Answer

y' +

\frac{(3 x + 2)}{x} y =

− e^{x}

4-217 §4.5 source ↗

Write the following first-order differential equations in standard form.

$\frac{d y}{d t} =$ $y x (x + 1)$

Answer

\frac{d y}{d t} -

y x (x + 1) =

0

4-219 §4.5 source ↗

What are the integrating factors for the following differential equations?

$y' + e^{x} y = sin x$

Answer

e^{e^{x}}

4-221 §4.5 source ↗

What are the integrating factors for the following differential equations?

$\frac{d y}{d x} =$ $tanh (x) y +$ $1$

Answer

− ln (cosh x)

4-223 §4.5 source ↗

Solve the following differential equations by using integrating factors.

$y' = 3 y + 2$

Answer

y = C e^{3 x} - \frac{2}{3}

4-225 §4.5 source ↗

Solve the following differential equations by using integrating factors.

$x y' = 3 y - 6 x^{2}$

Answer

y = C x^{3} + 6 x^{2}

4-227 §4.5 source ↗

Solve the following differential equations by using integrating factors.

$y' = 3 x + x y$

Answer

y = C e^{x^{2} / 2} - 3

4-229 §4.5 source ↗

Solve the following differential equations by using integrating factors.

$sin (x) y' =$ $y +$ $2 x$

Answer

y =

C tan (\frac{x}{2}) -

2 x +

4 tan (\frac{x}{2}) ln (sin (\frac{x}{2}))

4-231 §4.5 source ↗

Solve the following differential equations by using integrating factors.

$x y' = 3 y + x^{2}$

Answer

y = C x^{3} - x^{2}

4-233 §4.5 source ↗

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

[T] $(x + 2) y' =$ $2 y -$ $1$

Answer

y =

C {(x + 2)}^{2} +

\frac{1}{2}

4-235 §4.5 source ↗

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

[T] $x y' +$ $\frac{y}{2} =$ $sin (3 t)$

Answer

y =

\frac{C}{\sqrt{x}} +

2 sin (3 t)

4-237 §4.5 source ↗

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

[T] $(x + 1) y' =$ $3 y +$ $x^{2} +$ $2 x +$ $1$

Answer

y =

C {(x + 1)}^{3} -

x^{2} -

2 x -

1

4-239 §4.5 source ↗

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?

[T] $\sqrt{x^{2} + 1} y' = y + 2$

Answer

y =

C e^{{sinh}^{−1} x} -

2

4-241 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

$y' +$ $y =$ $x,$ $y (0) =$ $3$

Answer

y = x + 4 e^{–x} - 1

4-243 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

$x y' =$ $y -$ $3 x^{3},$ $y (1) =$ $0$

Answer

y =

-

\frac{3 x}{2} (x^{2} - 1)

4-245 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

$(1 + x^{2}) y' =$ $y -$ $1,$ $y (0) =$ $0$

Answer

y = 1 - e^{{tan}^{−1} x}

4-247 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

$(2 + x) y' =$ $y +$ $2 +$ $x,$ $y (0) =$ $0$

Answer

y =

(x + 2) ln (\frac{x + 2}{2})

4-249 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

$\sqrt{x} y' =$ $y +$ $2 x,$ $y (0) =$ $1$

Answer

y =

2 e^{2 \sqrt{x}} -

2 x -

2 \sqrt{x} -

1

4-251 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k .$ Set up the differential equation and solve for the velocity given an initial velocity of $0 .$

Answer

v (t) =

\frac{g m}{k} (1 - e^{− k t / m})

4-253 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

[T] Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall $5000$ meters if the mass is $100$ kilograms, the acceleration due to gravity is $9.8$ m/s² and the proportionality constant is $4 ?$

Answer

40.451

seconds

4-255 §4.5 source ↗

Solve the following initial-value problems by using integrating factors.

Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)

Answer

\sqrt{\frac{g m}{k}}

4-257 §4.5 source ↗

For the following problems, determine how parameter $a$ affects the solution.

Solve the generic equation $y' = a x + y .$ How does varying $a$ change the behavior?

Answer

y =

C e^{x} -

a (x + 1)

4-259 §4.5 source ↗

For the following problems, determine how parameter $a$ affects the solution.

Solve the generic equation $y' = a x + x y .$ How does varying $a$ change the behavior?

Answer

y = C e^{x^{2} / 2} - a

4-261 §4.5 source ↗

For the following problems, determine how parameter $a$ affects the solution.

Solve $y' - y = e^{k t}$ with the initial condition $y (0) = 0 .$ As $k$ approaches $1,$ what happens to your formula?

Answer

y = \frac{e^{k t} - e^{t}}{k - 1}

4-263 §4.R source ↗

True or False? Justify your answer with a proof or a counterexample.

The differential equation $y' = x - y$ is separable.

Answer

F

4-265 §4.R source ↗

True or False? Justify your answer with a proof or a counterexample.

You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

Answer

T

4-267 §4.R source ↗

For the following problems, find the general solution to the differential equations.

$y' =$ $2^{x} +$ ${cos}^{−1} x$

Answer

y (x) =

\frac{2^{x}}{ln (2)} +

x {cos}^{−1} x -

\sqrt{1 - x^{2}} +

C

4-269 §4.R source ↗

For the following problems, find the general solution to the differential equations.

$y' = e^{− y} sin x$

Answer

y (x) =

ln (C - cos x)

4-271 §4.R source ↗

For the following problems, find the general solution to the differential equations.

$y' = y ln y$

Answer

y (x) = e^{e^{C + x}}

4-273 §4.R source ↗

For the following problems, find the solution to the initial value problem.

$y' =$ $3 x -$ $cos x +$ $2,$ $y (0) =$ $4$

Answer

y (x) =

4 +

\frac{3}{2} x^{2} +

2 x -

sin x

4-275 §4.R source ↗

For the following problems, find the solution to the initial value problem.

$y' =$ $3 y^{2} (x + cos x),$ $y (0) =$ $−2$

Answer

y (x) =

-

\frac{2}{1 + 3 (x^{2} + 2 sin x)}

4-277 §4.R source ↗

For the following problems, find the solution to the initial value problem.

$y' =$ $3 y -$ $x +$ $6 x^{2},$ $y (0) =$ $−1$

Answer

y (x) =

−2 x^{2} -

2 x -

\frac{1}{3} -

\frac{2}{3} e^{3 x}

4-279 §4.R source ↗

For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.

$y' =$ $\frac{1}{x} +$ $ln x -$ $y,$ for $x > 0$

Answer

A direction field with arrows pointing up and to the right along a logarithmic curve that approaches negative infinity as x goes to zero and increases as x goes to infinity.

$y (x) =$ $C e^{− x} +$ $ln x$

4-281 §4.R source ↗

For the following problems, use Euler’s Method with $n = 5$ steps over the interval $t = [0, 1] .$ Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?

$y' =$ $3^{x} -$ $2 y,$ $y (0) =$ $0$

Answer

Euler:

0.6939,

exact solution:

y (x) =

\frac{3^{x} - e^{−2 x}}{2 + ln (3)}

4-283 §4.R source ↗

For the following problems, set up and solve the differential equations.

You throw a ball of mass $2$ kilograms into the air with an upward velocity of $8$ m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by $g = 9.8 {m/s}^{2} .$

Answer

\frac{40}{49}

second

4-285 §4.R source ↗

For the following problems, set up and solve the differential equations.

You drop the same ball of mass $5$ kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of $3$ and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

Answer

x (t) =

5000 +

\frac{245}{9} -

\frac{49}{3} t -

\frac{245}{9} e^{− 5 / 3 t},

t =

307.8

seconds

4-287 §4.R source ↗

For the following problems, set up and solve the differential equations.

A $1000$ -liter tank contains pure water and a solution of $0.2$ kg salt/L is pumped into the tank at a rate of $1$ L/min and is drained at the same rate. Solve for total amount of salt in the tank at time $t .$

Answer

T (t) =

200 (1 - e^{− t / 1000})

4-289 §4.R source ↗

For the following problems, set up and solve the differential equations.

The human population (in thousands) of Nevada in $1950$ was roughly $160 .$ If the carrying capacity is estimated at $10$ million individuals, and assuming a growth rate of $2 %$ per year, develop a logistic growth model and solve for the population in Nevada at any time (use $1950$ as time = 0). What population does your model predict for $2000 ?$ How close is your prediction to the true value of $1,998,257 ?$

Answer

P (t) =

\frac{1600000 e^{0.02 t}}{9840 + 160 e^{0.02 t}}