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Chapter 9

Differential Equations

Educators


Problem 1

Show that $y=x-x^{-1}$ is a solution of the differential equation $x y^{\prime}+y=2 x$

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Problem 2

Verify that $y=\sin x \cos x-\cos x$ is a solution of the initial-value problem
$$y^{\prime}+(\tan x) y=\cos ^{2} x \quad y(0)=-1$$
on the interval $-\pi / 2$ < $x$ < $\pi / 2$

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Problem 3

(a) For what values of $r$ does the function $y=e^{r x}$ satisfy the differential equation $2 y^{\prime \prime}+y^{\prime}-y=0 ?$

(b) If $r_{1}$ and $r_{2}$ are the values of $r$ that you found in part (a) show that every member of the family of functions $y=a e^{r_{1} x}+b e^{r_{2} x}$ is also a solution.

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Problem 4

(a) For what values of $k$ does the function $y=\cos k t$ satisfy the differential equation $4 y^{\prime \prime}=-25 y ?$

(b) For those values of $k,$ verify that every member of the family of functions $y=A \sin k t+B \cos k t$ is also a solution.

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Problem 5

Which of the following functions are solutions of the differential equation $y^{\prime \prime}+y=\sin x ?$
$$ \begin{array}{ll}{\text { (a) } y=\sin x} & {\text { (b) } y=\cos x} \\ {\text { (c) } y=\frac{1}{2} x \sin x} & {\text { (d) } y=-\frac{1}{2} x \cos x}\end{array} $$

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Problem 6

(a) Show that every member of the family of functions
$y=(\ln x+C) / x$ is a solution of the differential equation
$x^{2} y^{\prime}+x y=1$

(b) Illustrate part (a) by graphing several members of the
family of solutions on a common screen.

(c) Find a solution of the differential equation that satisfies
the initial condition $y(1)=2 .$

(d) Find a solution of the differential equation that satisfies
the initial condition $y(2)=1$

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Problem 7

(a) What can you say about a solution of the equation $y^{\prime}=-y^{2}$ just by looking at the differential equation?

(b) Verify that all members of the family $y=1 /(x+C)$ are
solutions of the equation in part (a).

(c) Can you think of a solution of the differential equation
$y^{\prime}=-y^{2}$ that is not a member of the family in part (b)

(d) Find a solution of the initial-value problem

$$y^{\prime}=-y^{2} \quad y(0)=0.5$$

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Problem 8

(a) What can you say about the graph of a solution of the equation $y=x y^{3}$ when $x$ is close to 0$?$ What if $x$ is large?

(b) Verify that all members of the family $y=\left(c-x^{2}\right)^{-1 / 2}$ are solutions of the differential equation $y^{\prime}=x y^{3}$

(c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)?

(d) Find a solution of the initial-value problem

$$y=x y^{3} \quad y(0)=2$$

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Problem 9

A population is modeled by the differential equation
$$\frac{d P}{d t}=1.2 P\left(1-\frac{P}{4200}\right)$$
(a) For what values of $P$ is the population increasing?
(b) For what values of $P$ is the population decreasing?
(c) What are the equilibrium solutions?

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Problem 10

A function $y(t)$ satisfies the differential equation
$$\frac{d y}{d t}=y^{4}-6 y^{3}+5 y^{2}$$
(a) What are the constant solutions of the equation?
(b) For what values of $y$ is $y$ increasing?
(c) For what values of $y$ is $y$ decreasing?

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Problem 11

Explain why the functions with the given graphs can't be solutions of the differential equation
$$\frac{d y}{d t}=e^{t}(y-1)^{2}$$

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Problem 12

The function with the given graph is a solution of one of the following differential equations. Decide which is the correct equation and justify your answer.
$$A. \quad y^{\prime}=1+x y \quad B. y^{\prime}=-2 x y \quad C. y^{\prime}=1-2 x y$$

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Problem 13

Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function $P(t),$ the performance of someone learning a skill as a function of the
training time $t .$ The derivative $d P / d t$ represents the rate at which performance improves.
(a) When do you think $P$ increases most rapidly? What happens to $d P / d t$ as $t$ increases? Explain.

(b) If $M$ is the maximum level of performance of which the learner is capable, explain why the differential equation
$$\frac{d P}{d t}=k(M-P)$$
k a positive constant is a reasonable model for learning.

(c) Make a rough sketch of a possible solution of this differential equation.

Eduardo M.
Numerade Educator

Problem 14

Suppose you have just poured a cup of freshly brewed coffee with temperature $95^{\circ} \mathrm{C}$ in a room where the temperature is $20^{\circ} \mathrm{C}$ .

(a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain.

(b) Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this
difference is not too large. Write a differential equation that expresses Newton's Law of Cooling for this particular situation. What is the initial condition? In view of your answer
to part (a), do you think this differential equation is an appropriate model for cooling?

(c) Make a rough sketch of the graph of the solution of the
initial-value problem in part (b).

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