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Calculus

James Stewart

Chapter 9

Differential Equations - all with Video Answers

Educators

Section 1

9.1 Exercises

01:27

Problem 1

Show that $y=\frac{2}{3} e^x+e^{-2 x}$ is a solution of the differential equation $y^{\prime}+2 y=2 e^x$.

Stephen Hobbs
Stephen Hobbs
Numerade Educator
01:37

Problem 2

Verify that $y=-t \cos t-t$ is a solution of the initial-value problem

$$
t \frac{d y}{d t}=y+t^2 \sin t \quad y(\pi)=0
$$

Anand Jangid
Anand Jangid
Numerade Educator
04:29

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 $y=a e^{n_x x}+b e^{r^x x}$ is also a solution.

WM
William Mead
Numerade Educator
05:17

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.

Madeline Zackeo
Madeline Zackeo
Numerade Educator
06:10

Problem 5

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

Madeline Zackeo
Madeline Zackeo
Numerade Educator
01:56

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 $y(2)=1$.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
05:45

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

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

MP
Mirjeta Pasha
Numerade Educator
06:58

Problem 8

(a) What can you say about the graph of a solution of the equation $y^{\prime}=x y^3$ 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

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

Madeline Zackeo
Madeline Zackeo
Numerade Educator
02:55

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?

Stephen Hobbs
Stephen Hobbs
Numerade Educator
01:00

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?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:32

Problem 11

Explain why the functions with the given graphs can't be solutions of the differential equation

$$
\frac{d y}{d t}=e^t(y-1)^2
$$

(b)

Madeline Zackeo
Madeline Zackeo
Numerade Educator
01:59

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. $y^{\prime}=1+x y$
B. $y^{\prime}=-2 x y$
C. $y^{\prime}=1-2 x y$

Stephen Hobbs
Stephen Hobbs
Numerade Educator
04:02

Problem 13

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

Madeline Zackeo
Madeline Zackeo
Numerade Educator
03:47

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).

Madeline Zackeo
Madeline Zackeo
Numerade Educator
02:41

Problem 15

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

$$
\frac{d P}{d t}=k(M-P) \quad k \text { a positive constant }
$$

is a reasonable model for learning.
(c) Make a rough sketch of a possible solution of this differential equation.

Eduardo Muntaner
Eduardo Muntaner
Numerade Educator