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Chapter 9

Differential Equations

Educators


Problem 1

$\begin{array}{l}{\text { Show that } y=\frac{2}{3} e^{x}+e^{-2 x} \text { is a solution of the differential }} \\ {\text { equation } v^{\prime}+2 y=2 e^{x}}\end{array}$

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Problem 2

$\begin{array}{c}{\text { Verify that } y=-t \cos t-t \text { is a solution of the initial-value }} \\ {\text { problem }} \\ {t \frac{d y}{d t}=y+t^{2} \sin t \quad y(\pi)=0}\end{array}$

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Problem 3

$\begin{array}{l}{\text { (a) For what values of } r \text { does the function } y=e^{r x} \text { satisfy the }} \\ {\text { differential equation } 2 y^{\prime \prime}+y^{\prime}-y=0 ?} \\ {\text { (b) If } r_{1} \text { and } r_{2} \text { are the values of } r \text { that you found in part (a) }} \\ {\text { show that every member of the family of functions }} \\ {y=a e^{n x}+b e^{n x} \text { is also a solution. }}\end{array}$

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Problem 4

$\begin{array}{l}{\text { (a) For what values of } k \text { does the function } y=\cos k t \text { satisfy }} \\ {\text { the differential equation } 4 y^{\prime \prime}=-25 y ?} \\ {\text { (b) For those values of } k, \text { verify that every member of the }} \\ {\text { family of functions } y=A \sin k t+B \cos k t \text { is also a }} \\ {\text { solution. }}\end{array}$

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Problem 5

Which of the following functions are solutions of the differ-
ential 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

$\begin{array}{l}{\text { (a) Show that every member of the family of functions }} \\ {y=(\ln x+C) / x \text { is a solution of the differential equation }} \\ {x^{2} y^{\prime}+x y=1}\end{array}$
$\begin{array}{l}{\text { (b) Illustrate part (a) by graphing several members of the }} \\ {\text { family of solutions on a common screen. }} \\ {\text { (c) Find a solution of the differential equation that satisfies }} \\ {\text { the initial condition } y(1)=2 .} \\ {\text { (d) Find a solution of the differential equation that satisfies }} \\ {\text { the initial condition } y(2)=1}\end{array}$

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Problem 7

$\begin{array}{l}{\text { (a) What can you say about a solution of the equation }} \\ {y^{\prime}=-y^{2} \text { just by looking at the differential equation? }} \\ {\text { (b) Verify that all members of the family } y=1 /(x+C) \text { are }} \\ {\text { solutions of the equation in part (a). }}\end{array}$
$\begin{array}{c}{\text { (c) Can you think of a solution of the differential equation }} \\ {y^{\prime}=-y^{2} \text { that is not a member of the family in part (b)? }} \\ {\text { (d) Find a solution of the initial-value problem }} \\ {y^{\prime}=-y^{2} \quad y(0)=0.5}\end{array}$

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Problem 8

$\begin{array}{l}{\text { (a) What can you say about the graph of a solution of the }} \\ {\text { equation } y^{\prime}=x y^{3} \text { when } x \text { is close to } 0 ? \text { What if } x \text { is }} \\ {\text { large? }} \\ {\text { (b) Verify that all members of the family } y=\left(c-x^{2}\right)^{-1 / 2}} \\ {\text { are solutions of the differential equation } y^{\prime}=x y^{3}}\end{array}$
$\begin{array}{c}{\text { (c) Graph several members of the family of solutions on a }} \\ {\text { common screen. Do the graphs confirm what you pre- }} \\ {\text { dicted in part (a)? }} \\ {\text { (d) Find a solution of the initial-value problem }} \\ {y^{\prime}=x y^{3} \quad y(0)=2}\end{array}$

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Problem 9

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

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Problem 10

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

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Problem 11

$\begin{array}{c}{\text { Explain why the functions with the given graphs can't be solu- }} \\ {\text { tions of the differential equation }} \\ {\frac{d y}{d t}=e^{t}(y-1)^{2}}\end{array}$

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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.
$\begin{array}{llll}{\text { A. } y^{\prime}=1+x y} & {\text { B. } y^{\prime}=-2 x y} & {\text { c. } y^{\prime}=1-2 x y}\end{array}$

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Problem 13

Match the differential equations with the solution graphs
labeled I-IV. Give reasons for your choices.
$\begin{array}{ll}{\text { (a) } y^{\prime}=1+x^{2}+y^{2}} & {\text { (b) } y^{\prime}=x e^{-x^{2}-y^{2}}} \\ {\text { (c) } y^{\prime}=\frac{1}{1+e^{x^{2}+y^{2}}}} & {\text { (d) } y^{\prime}=\sin (x y) \cos (x y)}\end{array}$

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Problem 14

$\begin{array}{l}{\text { Suppose you have just poured a cup of freshly brewed coffee }} \\ {\text { with temperature } 95^{\circ} \mathrm{C} \text { in a room where the temperature }} \\ {\text { is } 20^{\circ} \mathrm{C} .} \\ {\text { (a) When do you think the coffee cools most quickly? What }} \\ {\text { happens to the rate of cooling as time goes by? Explain. }}\end{array}$
$\begin{array}{l}{\text { (b) Newton's Law of Cooling states that the rate of cooling }} \\ {\text { of an object is proportional to the temperature difference }} \\ {\text { between the object and its surroundings, provided that this }} \\ {\text { difference is not too large. Write a differential equation that}}\end{array}$
$\begin{array}{l}{\text { expresses Newton s Law of Cooling for this particular situ- }} \\ {\text { ation. What is the initial condition? In view of your answer }} \\ {\text { to part (a), do you think this differential equation is an }} \\ {\text { appropriate model for cooling? }}\end{array}$
$\begin{array}{l}{\text { (c) Make a rough sketch of the graph of the solution of the }} \\ {\text { initial-value problem in part (b). }}\end{array}$

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Problem 15

$\begin{array}{l}{\text { Psychologists interested in learning theory study learning }} \\ {\text { curves. A learning curve is the graph of a function } P(t), \text { the }} \\ {\text { performance of someone learning a skill as a function of the }} \\ {\text { training time } t . \text { The derivative } d P / d t \text { represents the rate at }} \\ {\text { which performance improves. }}\end{array}$
$\begin{array}{c}{\text { (a) When do you think } P \text { increases most rapidly? What }} \\ {\text { happens to } d P / d t \text { as } t \text { increases? Explain. }} \\ {\text { (b) If } M \text { is the maximum level of performance of which the }} \\ {\text { learner is capable, explain why the differential equation }} \\ {\frac{d P}{d t}=k(M-P) \quad k \text { a positive constant }}\end{array}$
$\begin{array}{l}{\text { is a reasonable model for learning. }} \\ {\text { (c) Make a rough sketch of a possible solution of this differen- }} \\ {\text { tial equation. }}\end{array}$

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