Question 1

Choose the differential equation corresponding to this direction field.

Accepted Answer

A)  $y ^ { \prime } = x + y - 1$ 
B)  $y ^ { \prime } = \sin x \sin y$ 
C)  $y ^ { \prime } = y + x y$ 
D)  $y ^ { \prime } = 2 - y$ 
E)  $y ^ { \prime } = x ( 2 - y )$ 
A)  $y ^ { \prime } = x + y - 1$ 
B)  $y ^ { \prime } = \sin x \sin y$ 
C)  $y ^ { \prime } = y + x y$ 
D)  $y ^ { \prime } = 2 - y$ 
E)  $y ^ { \prime } = x ( 2 - y )$

Question 2

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \cos x , y ( 3 \pi ) = 0$

Accepted Answer

The answer of Solve the initial-value problem. \(x y ^...

Question 3

Solve the differential equation. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

Accepted Answer

A)  $u = \frac { t ^ { 2 } + C } { t + 6 }$ 
B)  $u = 6 t + \frac { t ^ { 2 } } { 6 } + C$ 
C)  $u = \frac { 6 t + t ^ { 2 } + C } { t + 6 }$ 
D)  $u = \frac { 6 t + \frac { t ^ { 2 } } { 2 } + C } { t + 6 }$ 
E)  $u = \frac { 6 t + \frac { 1 } { 2 } t ^ { 2 } } { \frac { 3 } { 6 } t + 2 } + C$ 
A)  $u = \frac { t ^ { 2 } + C } { t + 6 }$ 
B)  $u = 6 t + \frac { t ^ { 2 } } { 6 } + C$ 
C)  $u = \frac { 6 t + t ^ { 2 } + C } { t + 6 }$ 
D)  $u = \frac { 6 t + \frac { t ^ { 2 } } { 2 } + C } { t + 6 }$ 
E)  $u = \frac { 6 t + \frac { 1 } { 2 } t ^ { 2 } } { \frac { 3 } { 6 } t + 2 } + C$

Question 4

Let $P ( t )$ be the performance level of someone learning a skill as a function of the training time t. The graph of P is called a learning curve. We propose the differential equation $\frac { d P } { d t } = r ( G - P ( t ) )$ as a reasonable model for learning, where r is a positive constant. Solve it as a linear differential equation.

Accepted Answer

A)  $P ( t ) = G - C e ^ { - r t }$ 
B)  $P ( t ) = G + C e ^ { - r t }$ 
C)  $P ( t ) = G + C e ^ { rt }$ 
D)  $P ( t ) = G C e ^ { n t }$ 
E)  $P ( t ) = G - C e ^ { rt }$ 
A)  $P ( t ) = G - C e ^ { - r t }$ 
B)  $P ( t ) = G + C e ^ { - r t }$ 
C)  $P ( t ) = G + C e ^ { rt }$ 
D)  $P ( t ) = G C e ^ { n t }$ 
E)  $P ( t ) = G - C e ^ { rt }$

Question 5

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of P is the population increasing?

Accepted Answer

A) $P > 1.4$ B) $P > 4560$ C) $P < 4580$ D) $0 < P < 4560$ E) $0 < P < 1.4$ A) $P > 1.4$ B) $P > 4560$ C) $P < 4580$ D) $0 < P < 4560$ E) $0 < P < 1.4$

Question 6

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Accepted Answer

A)  $y = C e ^ { - 2 / x }$ 
B)  $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$ 
C)  $y = C x e ^ { 2 x }$ 
D)  $y = e ^ { 1 / x } x + C e ^ { 1 / x }$ 
E)  $\text { none of these }$ 
A)  $y = C e ^ { - 2 / x }$ 
B)  $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$ 
C)  $y = C x e ^ { 2 x }$ 
D)  $y = e ^ { 1 / x } x + C e ^ { 1 / x }$ 
E)  $\text { none of these }$

Question 7

Solve the initial-value problem. $r ^ { \prime } + 2 t r = r , \quad r ( 0 ) = 2$

Accepted Answer

A)  $r ( t ) = 2 e ^ { t - t ^ { 2 } }$ 
B)  $r ( t ) = 2 e ^ { t - 2 t }$ 
C)  $r ( t ) = e ^ { t - 2 t ^ { 2 } }$ 
D)  $r ( t ) = \frac { e ^ { t ^ { 2 } } } { 2 }$ 
E)  $r ( t ) = 2 e ^ { 2 t }$ 
A)  $r ( t ) = 2 e ^ { t - t ^ { 2 } }$ 
B)  $r ( t ) = 2 e ^ { t - 2 t }$ 
C)  $r ( t ) = e ^ { t - 2 t ^ { 2 } }$ 
D)  $r ( t ) = \frac { e ^ { t ^ { 2 } } } { 2 }$ 
E)  $r ( t ) = 2 e ^ { 2 t }$

Question 8

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

Accepted Answer

The answer of Solve the differential equation. \(\frac { d...

Question 9

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 2 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Accepted Answer

A)  $y = \ln ( 2 - \cos x ) ( 2 + \cos x )$ 
B)  $y = \frac { ( 7 - \cos x ) } { ( 2 + \cos x ) }$ 
C)  $y = \ln \frac { \cos x } { ( 2 + \cos x ) }$ 
D)  $y = \frac { \ln ( 2 + \cos x ) } { \ln ( 2 - \cos x ) }$ 
E)  $y = \ln ( 7 - \cos x ) - \ln ( 1 + \cos x )$ 
A)  $y = \ln ( 2 - \cos x ) ( 2 + \cos x )$ 
B)  $y = \frac { ( 7 - \cos x ) } { ( 2 + \cos x ) }$ 
C)  $y = \ln \frac { \cos x } { ( 2 + \cos x ) }$ 
D)  $y = \frac { \ln ( 2 + \cos x ) } { \ln ( 2 - \cos x ) }$ 
E)  $y = \ln ( 7 - \cos x ) - \ln ( 1 + \cos x )$

Question 10

Solve the differential equation. $( 7 + t ) \frac { d u } { d t } + u = 7 + t , t > 0$

Accepted Answer

The answer of Solve the differential equation. \(( 7 +...

Question 11

Which equation does the function $y = e ^ { - 6 t }$ satisfy?

Accepted Answer

A)  $y ^ { \prime \prime } + y ^ { \prime } + 42 y = 0$ 
B)  $y ^ { \prime \prime } - 3 y ^ { \prime } + 42 y = 0$ 
C)  $y ^ { \prime \prime } - y ^ { \prime } - 42 y = 0$ 
D)  $y ^ { \prime \prime } + y ^ { \prime } - 42 y = 0$ 
E)  $y ^ { \prime \prime } - y ^ { \prime } + 42 y = 0$ 
A)  $y ^ { \prime \prime } + y ^ { \prime } + 42 y = 0$ 
B)  $y ^ { \prime \prime } - 3 y ^ { \prime } + 42 y = 0$ 
C)  $y ^ { \prime \prime } - y ^ { \prime } - 42 y = 0$ 
D)  $y ^ { \prime \prime } + y ^ { \prime } - 42 y = 0$ 
E)  $y ^ { \prime \prime } - y ^ { \prime } + 42 y = 0$

Question 12

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of $5000$ inhabitants, $130$ people have a disease at the beginning of the week and $1500$ have it at the end of the week. How long does it take for $50 \%$ of the population to be infected?

Accepted Answer

The answer of One model for the spread of an...

Question 13

Experiments show that if the chemical reaction $\mathrm { N } _ { 2 } \mathrm { O } _ { 5 } \rightarrow 2 \mathrm { NO } _ { 2 } + \frac { 1 } { 2 } \mathrm { O } _ { 2 }$ takes place at $45 ^ { \circ } \mathrm { C }$ , the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows : $- \frac { d \left[ \mathrm {~N} _ { 2 } \mathrm { O } _ { 5 } \right] } { d t } = 0.0007 \left[ \mathrm {~N} _ { 2 } \mathrm { O } _ { 5 } \right]$ How long will the reaction take to reduce the concentration of $\mathrm { N } _ { 2 } \mathrm { O } _ { 5 }$ to 50% of its original value?

Accepted Answer

The answer of Experiments show that if the chemical reaction...

Question 14

Let c be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where k is a positive constant, is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 106 }$ . If $5$ such rabbits breed initially and the warren has $25$ rabbits after $5$ months, then when is doomsday?

Accepted Answer

The answer of Let c be a positive number. A...

Question 15

A sum of $\$ 2170$ is invested at $11 \%$ interest. If $A ( t )$ is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Accepted Answer

The answer of A sum of $\$ 2170$ is invested...

Question 16

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P provided that the temperature is constant. At $15 ^ { \circ } \mathrm { C }$ the pressure is $100.5 \mathrm { kPa }$ at sea level and $82.38 \mathrm { kPa }$ at $h = 1,070$ . What is the pressure at an altitude of $1,000 \mathrm {~m}$ ?

Accepted Answer

The answer of The rate of change of atmospheric pressure...

Question 17

An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If $s ( t )$ is the distance dropped after t seconds, then the speed is $v = s ^ { \prime } ( t )$ and the acceleration is $a = v ^ { \prime } ( t )$ . If g is the acceleration due to gravity, then the downward force on the object is $m g - c v$ , where c is a positive constant, and Newton's Second Law gives $m \frac { d v } { d t } = m g - c v$ .
Find the limiting velocity.

Accepted Answer

The answer of An object with mass m is dropped...

Question 18

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

Accepted Answer

A)  $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$ 
B)  $y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$ 
C)  $y = C e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + C x$ 
E)  $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$ 
A)  $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$ 
B)  $y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$ 
C)  $y = C e ^ { - \sin x }$ 
D)  $y = e ^ { - \sin x } + C x$ 
E)  $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$

Question 19

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)

Accepted Answer

A)  78.3 billion 
B)  24.1 billion 
C)  17.1 billion 
D)  59.2 billion 
E)  32.9 billion 
A)  78.3 billion 
B)  24.1 billion 
C)  17.1 billion 
D)  59.2 billion 
E)  32.9 billion

Question 20

For what values of k does the function $y = \cos k t$ satisfy the differential equation $9 y ^ { \prime \prime } = - 25 y$ ?

Accepted Answer

A)  $k = - \frac { 1 } { 5 }$ 
B)  $k = \frac { 1 } { 5 }$ 
C)  $k = \frac { 1 } { 7 }$ 
D)  $k = - \frac { 5 } { 7 }$ 
E)  $k = \frac { 5 } { 7 }$ 
A)  $k = - \frac { 1 } { 5 }$ 
B)  $k = \frac { 1 } { 5 }$ 
C)  $k = \frac { 1 } { 7 }$ 
D)  $k = - \frac { 5 } { 7 }$ 
E)  $k = \frac { 5 } { 7 }$

Choose the differential equation corresponding to this direction field.

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \cos x , y ( 3 \pi ) = 0$

Solve the differential equation. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of P is the population increasing?

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the initial-value problem. $r ^ { \prime } + 2 t r = r , \quad r ( 0 ) = 2$

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 2 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Solve the differential equation. $( 7 + t ) \frac { d u } { d t } + u = 7 + t , t > 0$

Which equation does the function $y = e ^ { - 6 t }$ satisfy?

A sum of $\$ 2170$ is invested at $11 \%$ interest. If $A ( t )$ is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

For what values of k does the function $y = \cos k t$ satisfy the differential equation $9 y ^ { \prime \prime } = - 25 y$ ?

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 9: Differential Equations

Choose the differential equation corresponding to this direction field.

Solve the initial-value problem. xy′=y+x2cos⁡x,y(3π)=0x y ^ { \prime } = y + x ^ { 2 } \cos x , y ( 3 \pi ) = 0xy′=y+x2cosx,y(3π)=0

Solve the differential equation. (6+t)dudt+u=6+t,t>0( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0(6+t)dtdu​+u=6+t,t>0

A population is modeled by the differential equation. dPdt=1.4P(1−P4560)\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)dtdP​=1.4P(1−4560P​) For what values of P is the population increasing?

Solve the differential equation. x2dydx−y=2x3e−1/xx ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }x2dxdy​−y=2x3e−1/x

Solve the initial-value problem. r′+2tr=r,r(0)=2r ^ { \prime } + 2 t r = r , \quad r ( 0 ) = 2r′+2tr=r,r(0)=2

Solve the differential equation. dudt=22+2u+11t+ut\frac { d u } { d t } = 22 + 2 u + 11 t + u tdtdu​=22+2u+11t+ut

Solve the initial-value problem. (1+cos⁡x)dydx=(2+e−y)sin⁡x,y(0)=0( 1 + \cos x ) \frac { d y } { d x } = \left( 2 + e ^ { - y } \right) \sin x , y ( 0 ) = 0(1+cosx)dxdy​=(2+e−y)sinx,y(0)=0

Solve the differential equation. (7+t)dudt+u=7+t,t>0( 7 + t ) \frac { d u } { d t } + u = 7 + t , t > 0(7+t)dtdu​+u=7+t,t>0

Which equation does the function y=e−6ty = e ^ { - 6 t }y=e−6t satisfy?

A sum of $2170\$ 2170$2170 is invested at 11%11 \%11% interest. If A(t)A ( t )A(t) is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(t)A ( t )A(t) .

Solve the differential equation. y′=xe−sin⁡x−ycos⁡xy ^ { \prime } = x e ^ { - \sin x } - y \cos xy′=xe−sinx−ycosx

For what values of k does the function y=cos⁡kty = \cos k ty=coskt satisfy the differential equation 9y′′=−25y9 y ^ { \prime \prime } = - 25 y9y′′=−25y ?

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \cos x , y ( 3 \pi ) = 0$

Solve the differential equation. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of P is the population increasing?

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the initial-value problem. $r ^ { \prime } + 2 t r = r , \quad r ( 0 ) = 2$

Solve the differential equation. $\frac { d u } { d t } = 22 + 2 u + 11 t + u t$

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 2 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

Solve the differential equation. $( 7 + t ) \frac { d u } { d t } + u = 7 + t , t > 0$

Which equation does the function $y = e ^ { - 6 t }$ satisfy?

A sum of $\$ 2170$ is invested at $11 \%$ interest. If $A ( t )$ is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

For what values of k does the function $y = \cos k t$ satisfy the differential equation $9 y ^ { \prime \prime } = - 25 y$ ?