Question 1

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

Accepted Answer

A)  $125 ^ { \circ } \mathrm { F }$ 
B)  $100 ^ { \circ } \mathrm { F }$ 
C)  $65 ^ { \circ } \mathrm { F }$ 
D)  $50 ^ { \circ } \mathrm { F }$ 
E)  $0 ^ { \circ } \mathrm { F }$ 
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 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

A)  centimeters per second 
B)  feet per second 
C)  feet per second per second 
D)  kilograms per centimeter 
E)  kilograms per centimeter per second 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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)  $\frac { d A } { d t } = 15 + .05 A$ 
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 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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)  $\frac { d T } { d t } = T ( T - 65 )$ 
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 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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)  $\frac { d P } { d t } = k P ^ { 2 } , P ( 0 ) = 5000$ 
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 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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)  $\frac { d T } { d t } = T ( T - 70 )$ 
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)  $\frac { d T } { d t } = T ( T - 70 )$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 1: Introduction to Differential Equations

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

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

The differential equation y′′+2y′+3y=sin⁡yy ^ { \prime \prime } + 2 y ^ { \prime } + 3 y = \sin yy′′+2y′+3y=siny is

The solution of the initial value problem y′=3y,y(0)=2y ^ { \prime } = 3 y , y ( 0 ) = 2y′=3y,y(0)=2 is y=ce3xy = c e ^ { 3 x }y=ce3x , where c=c =c=

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

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

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

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

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

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

The values of m for which y=xmy = x ^ { m }y=xm is a solution of x2y′′−5xy′+8y=0x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 8 y = 0x2y′′−5xy′+8y=0 are

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)P ( t )P(t) , as a function of time, t, is

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

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

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

The initial value problem y′=y2−16,y(x0)=y0y ^ { \prime } = \sqrt { y ^ { 2 } - 16 } , y \left( x _ { 0 } \right) = y _ { 0 }y′=y2−16​,y(x0​)=y0​ has a unique solution guaranteed by Theorem 1.1 if

The differential equation y′+3y=sin⁡xy ^ { \prime } + 3 y = \sin xy′+3y=sinx is

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

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

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

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

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

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

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

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

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

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

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