Deck 1: Introduction to Differential Equations

Question 1

The solution of the initial value problem $y ^ { \prime } = 2 y + x , y ( - 1 ) = 1 / 2$ is $y = - x / 2 - 1 / 4 + c e ^ { 2 x }$ , where $c =$

A) 2
B) $e ^ { 2 } / 4$
C) $e ^ { 2 }$
D) $e ^ { 2 } / 2$
E) 1

Answer 1

$e ^ { 2 } / 4$

Question 2

The initial value problem $y ^ { \prime } = \sqrt { y ^ { 2 } - 16 } , y \left( x _ { 0 } \right) = y _ { 0 }$ has a unique solution guaranteed by Theorem 1.1 if

A) $y _ { 0 } = 4$
B) $y _ { 0 } = - 4$
C) $y _ { 0 } = 0$
D) $y _ { 0 } = 8$
E) $y _ { 0 } = 1$

Answer 2

$y _ { 0 } = 8$

Question 3

In the previous problem, over a long period of time, the total amount of salt in the tank will approach

A) 300 pounds
B) 500 pounds
C) 1000 pounds
D) 3000 pounds
E) 5000 pounds

Answer 3

D

Question 4

The temperature of a cup of coffee obeys Newton's law of cooling. The initial temperature of the coffee is $140 ^ { \circ } \mathrm { F }$ and one minute later, it is $125 ^ { \circ } \mathrm { F }$ . The ambient temperature of the room is $65 ^ { \circ } \mathrm { F }$ . If $T ( t )$ represents the temperature of the coffee at time t, the correct differential equation for the temperature is

A) $\frac { d T } { d t } = k ( T - 125 )$
B) $\frac { d T } { d t } = k ( T - 140 )$
C) $\frac { d T } { d t } = k ( T - 65 )$
D) $\frac { d T } { d t } = T ( T - 140 )$
E) $\frac { d T } { d t } = T ( T - 65 )$

Question 5

A large mixing tank initially contains 1000 gallons of water in which 40 pounds of salt have been dissolved. Another brine solution is pumped into the tank at the rate of 5 gallons per minute, and the resulting mixture is pumped out at the same rate. The concentration of the incoming brine solution is 3 pounds of salt per gallon. If $A ( t )$ represents the amount of salt in the tank at time t, the correct differential equation for A is

A) $\frac { d A } { d t } = 3 - .005 A$
B) $\frac { d A } { d t } = 5 - .05 A$
C) $\frac { d A } { d t } = 15 - .005 A$
D) $\frac { d A } { d t } = 3 - .05 A$
E) $\frac { d A } { d t } = 15 + .05 A$

Question 6

The differential equation $y ^ { \prime \prime \prime } + 2 y ^ { \prime \prime } + 3 y ^ { \prime } + 7 y = 0$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 7

In the LRC circuit problem in the text, the units for C, are

A) ohms
B) farads
C) amperes
D) henrys
E) coulombs

Question 8

The differential equation $y ^ { \prime } + 3 y = \sin x$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 9

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 6 y ^ { \prime } - 7 y = 0$ are

A) 1 and 7
B) $- 1$ and 6
C) 1 and 6
D) 1 and $- 6$
E) $- 1$ and 7

Question 10

The values of m for which $y = e ^ { xxx }$ is a solution of $y ^ { \prime \prime } - 9 y ^ { \prime } + 20 y = 0$ are

A) 4 and $- 5$
B) $- 4$ and $- 5$
C) 3 and 6
D) 4 and 5
E) 3 and 5

Question 11

In the previous problem, after a long period of time, the temperature of the coffee approaches

A) $125 ^ { \circ } \mathrm { F }$
B) $100 ^ { \circ } \mathrm { F }$
C) $65 ^ { \circ } \mathrm { F }$
D) $50 ^ { \circ } \mathrm { F }$
E) $0 ^ { \circ } \mathrm { F }$

Question 12

The values of m for which $y = x ^ { m }$ is a solution of $x ^ { 2 } y ^ { \prime \prime } - 7 x y ^ { \prime } + 12 y = 0$ are

A) $- 3$ and 4
B) $- 2$ and $- 6$
C) 3 and 4
D) 2 and 6
E) 3 and $- 4$

Question 13

The population of a town increases at a rate proportional to its population. Its initial population is 5000. The correct initial value problem for the population, $P ( t )$ , as a function of time, t, is

A) $\frac { d P } { d t } = k P , P ( 0 ) = 5000$
B) $\frac { d P } { d t } = k P ^ { 2 } , P ( 0 ) = 500$
C) $\frac { d P } { d t } = k P , P ( 0 ) = 500$
D) $\frac { d P } { d t } = k P ( 1 - P ) , P ( 0 ) = 5000$
E) $\frac { d P } { d t } = k P ^ { 2 } , P ( 0 ) = 5000$

Question 14

The values of c for which $y = c$ is a constant solution of $y ^ { \prime } = y ^ { 2 } + 5 y - 6$ are

A) 1 and 6
B) $- 1$ and 6
C) 1 and $- 6$
D) $- 2$ and 3
E) 2 and 3

Question 15

The differential equation $y ^ { \prime \prime } + 2 y ^ { \prime } + 3 y = \sin y$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 16

In the falling body problem, the units of acceleration might be

A) centimeters per second
B) feet per second
C) feet per second per second
D) kilograms per centimeter
E) kilograms per centimeter per second

Question 17

In the LRC circuit problem in the text, R stands for

A) capacitance
B) resistance
C) current
D) inductance
E) charge on the capacitor

Question 18

The differential equation $y ^ { \prime \prime } + 2 y y ^ { \prime } + 3 y = 0$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 19

The differential equation $y ^ { \prime \prime \prime } + 2 y ^ { \prime \prime } + 3 x y ^ { \prime } - 4 e ^ { x } y = \sin x$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 20

The solution of the initial value problem $y ^ { \prime } = 5 y , y ( 1 ) = 3$ is $y = c e ^ { 5 x }$ , where $c =$

A) $3 e ^ { - 5 }$
B) 3
C) $3 e ^ { 5 }$
D) $- 3 e ^ { 5 }$
E) $- 3$

Question 21

The differential equation $y ^ { \prime \prime } + 2 y y ^ { \prime } + 3 y = 0$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 22

The values of m for which $y = x ^ { m }$ is a solution of $x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 8 y = 0$ are

A) 2 and 4
B) $- 2$ and $- 4$
C) 3 and 5
D) 2 and 3
E) 1 and 5

Question 23

The temperature of a cup of coffee obeys Newton's law of cooling. The initial temperature of the coffee is $150 ^ { \circ } \mathrm { F }$ and one minute later, it is $135 ^ { \circ } \mathrm { F }$ . The ambient temperature of the room is $70 ^ { \circ } \mathrm { F }$ . If $T ( t )$ represents the temperature of the coffee at time t, the correct differential equation for the temperature with side conditions is

A) $\frac { d T } { d t } = k ( T - 135 )$
B) $\frac { d T } { d t } = k ( T - 150 )$
C) $\frac { d T } { d t } = k ( T - 70 )$
D) $\frac { d T } { d t } = T ( T - 150 )$
E) $\frac { d T } { d t } = T ( T - 70 )$

Question 24

The population of a town increases at a rate proportional to its population. Its initial population is 1000. The correct initial value problem for the population, $P ( t )$ , as a function of time, t, is

A) $\frac { d P } { d t } = k P , P ( 0 ) = 1000$
B) $\frac { d P } { d t } = k P ^ { 2 } , P ( 0 ) = 100$
C) $\frac { d P } { d t } = k P , P ( 0 ) = 100$
D) $\frac { d P } { d t } = k P ( 1 - P ) , P ( 0 ) = 100$
E) $\frac { d P } { d t } = k P ^ { 2 } , P ( 0 ) = 1000$

Question 25

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ are

A) 2 and 4
B) $- 2$ and $- 3$
C) 3 and 4
D) 2 and 3
E) 1 and 5

Question 26

In the LRC circuit problem in the text, C stands for

A) capacitance
B) resistance
C) current
D) inductance
E) charge on the capacitor

Question 27

The solution of the initial value problem $y ^ { \prime } = 3 y , y ( 0 ) = 2$ is $y = c e ^ { 3 x }$ , where $c =$

A) 2
B) $- 2$
C) 3
D) $- 3$
E) 1

Question 28

The values of c for which $y = c$ is a constant solution of $y ^ { \prime } = y ^ { 2 } + 3 y - 4$ are

A) 1 and 4
B) $- 1$ and $- 3$
C) 1 and $- 4$
D) $- 1$ and 3
E) 1 and 3

Question 29

The differential equation $y ^ { \prime \prime \prime } + 2 y ^ { \prime \prime } + 3 x y ^ { \prime } - 4 e ^ { x } y = \sin x$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 30

In the LRC circuit problem in the text, the units of inductance, L, are

A) ohms
B) farads
C) amperes
D) henrys
E) coulombs

Question 31

The solution of the initial value problem $y ^ { \prime } = 2 y + x , y ( 1 ) = 1 / 4$ is $y = - x / 2 - 1 / 4 + c e ^ { 2 x }$ , where $c =$

A) 2
B) $e ^ { - 2 }$
C) $e ^ { - 1 }$
D) $e ^ { - 2 } / 2$
E) 1

Question 32

The values of m for which $y = e ^ { m x }$ is a solution of $y ^ { \prime \prime } - 4 y ^ { \prime } - 5 y = 0$ are

A) 1 and 4
B) $- 1$ and 4
C) 2 and 3
D) $- 2$ and $- 3$
E) $- 1$ and 5

Question 33

In the previous problem, after a long period of time, the temperature of the coffee approaches

A) $120 ^ { \circ } \mathrm { F }$
B) $100 ^ { \circ } \mathrm { F }$
C) $70 ^ { \circ } \mathrm { F }$
D) $65 ^ { \circ } \mathrm { F }$
E) $0 ^ { \circ } \mathrm { F }$

Question 34

The differential equation $y ^ { \prime \prime } + 2 y ^ { \prime } + 3 y = \sin y$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 35

The differential equation $y ^ { \prime \prime } + 2 y ^ { \prime } + 3 y = 0$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear

Question 36

In the previous problem, over a long period of time, the total amount of salt in the tank will approach

A) 30 pounds
B) 50 pounds
C) 100 pounds
D) 200 pounds
E) 300 pounds

Question 37

In the falling body problem, the units of acceleration might be

A) meters per second
B) feet per second
C) meters per second per second
D) kilograms per meter
E) kilograms per meter per second

Question 38

A large mixing tank initially contains 100 gallons of water in which 30 pounds of salt have been dissolved. Another brine solution is pumped into the tank at the rate of 4 gallons per minute, and the resulting mixture is pumped out at the same rate. The concentration of the incoming brine solution is 2 pounds of salt per gallon. If $A ( t )$ represents the amount of salt in the tank at time t, the correct differential equation for A is

A) $\frac { d A } { d t } = 8 - .02 A$
B) $\frac { d A } { d t } = 8 - .04 A$
C) $\frac { d A } { d t } = 4 - .04 A$
D) $\frac { d A } { d t } = 2 - .04 A$
E) $\frac { d A } { d t } = 4 - .08 A$

Question 39

The initial value problem $y ^ { \prime } = \sqrt { y ^ { 2 } - 9 } , y \left( x _ { 0 } \right) = y _ { 0 }$ has a unique solution guaranteed by Theorem 1.1 if

A) $y _ { 0 } = 3$
B) $y _ { 0 } = - 3$
C) $y _ { 0 } = 5$
D) $y _ { 0 } = 0$
E) $y _ { 0 } = 1$

Question 40

The differential equation $y ^ { \prime } + 3 y = \sin x$ is

A) first order linear
B) second order linear
C) third order linear
D) first order nonlinear
E) second order nonlinear