Question 1

Solve the differential equation.
$$8 y ^ { t t } + y ^ { t } = 0$$

Accepted Answer

The answer of Solve the differential equation.
\[8 y ^ {...

Question 2

Solve the differential equation.
$$y ^ { tt } + 10 y ^ { t } + 41 y = 0$$

Accepted Answer

The answer of Solve the differential equation.
\[y ^ { tt...

Question 3

Solve the initial-value problem. Select the correct answer.
$$y ^ { tt} - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 8$$

Accepted Answer

A)  
$y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$ 
B) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
C)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
D) 
$y ( x ) = \frac { 24 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 16 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
E) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$ 
A)  
$y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$ 
B) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
C)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
D) 
$y ( x ) = \frac { 24 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 16 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
E) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$

Question 4

A series circuit consists of a resistor $R = 32 \Omega$, an inductor with $L = 4 H$, a capacitor with $C = 0.003125 \mathrm {~F}$, and a $24 - \mathrm { V }$ battery. If the initial charge is $0.0008 \mathrm { C }$ and the initial current is 0 , find the current $I ( t )$ at time $t$.

Accepted Answer

The answer of A series circuit consists of a resistor...

Question 5

A spring with a $16 - \mathrm { kg }$ mass has natural length $0.8 \mathrm {~m}$ and is maintained stretched to a length of $1.2$ $\mathrm { m }$ by a force of $19.6 \mathrm {~N}$. If the spring is compressed to a length of $0.4 \mathrm {~m}$ and then released with zero velocity, find the position $x ( t )$ of the mass at any time $t$. Select the correct answer.

Accepted Answer

A)  $x ( t ) = 0.8 \sin ( 1.75 t )$ 
B)  $x ( t ) = - 0.4 \cos ( 1.75 t )$ 
C)  $x ( t ) = - 0.4 e ^ { - 4 t } \cos ( 0.4 t )$ 
D)  $x ( t ) = - 0.4 e ^ { - 4 t } \cos ( 1.75 t )$ 
E)  $x ( t ) = - 0.4 \cos ( 1.75 t ) + 0.4 \sin ( 1.75 t )$ 
A)  $x ( t ) = 0.8 \sin ( 1.75 t )$ 
B)  $x ( t ) = - 0.4 \cos ( 1.75 t )$ 
C)  $x ( t ) = - 0.4 e ^ { - 4 t } \cos ( 0.4 t )$ 
D)  $x ( t ) = - 0.4 e ^ { - 4 t } \cos ( 1.75 t )$ 
E)  $x ( t ) = - 0.4 \cos ( 1.75 t ) + 0.4 \sin ( 1.75 t )$

Question 6

Solve the differential equation.
$$y ^ { tt } + 4 y ^ { t} - 5 y = 0$$

Accepted Answer

A)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 5 x }$ 
B) $y = \frac { C _ { 1 } e ^ { x } } { 3 } + \frac { C _ { 2 } e ^ { - 5 x } } { 2 }$ 
C)  $y = C _ { 1 } e ^ { - 5 x } + C _ { 2 } e ^ { - 5 x }$ 
D)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$ 
A)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 5 x }$ 
B) $y = \frac { C _ { 1 } e ^ { x } } { 3 } + \frac { C _ { 2 } e ^ { - 5 x } } { 2 }$ 
C)  $y = C _ { 1 } e ^ { - 5 x } + C _ { 2 } e ^ { - 5 x }$ 
D)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$

Question 7

Solve the initial-value problem.
$$y ^ {tt } + 3 y ^ { t } - 4 y = 0 , y ( 0 ) = 2 , y ^ { t } ( 0 ) = 1$$

Accepted Answer

A) 
$y = e ^ { 2 x } + \frac { 1 } { 5 } e ^ { - 2 x }$ 
B) 
$y = \frac { 9 } { 5 } e ^ { x } + \frac { 1 } { 5 } e ^ { - 4 x }$ 
C) 
$y = \frac { 1 } { 5 } e ^ { 2 x } + \frac { 9 } { 5 } e ^ { - 2 x }$ 
D) 
$y = e ^ { x } + \frac { 1 } { 5 } e ^ { - 3 x }$ 
E)  None of these 
A) 
$y = e ^ { 2 x } + \frac { 1 } { 5 } e ^ { - 2 x }$ 
B) 
$y = \frac { 9 } { 5 } e ^ { x } + \frac { 1 } { 5 } e ^ { - 4 x }$ 
C) 
$y = \frac { 1 } { 5 } e ^ { 2 x } + \frac { 9 } { 5 } e ^ { - 2 x }$ 
D) 
$y = e ^ { x } + \frac { 1 } { 5 } e ^ { - 3 x }$ 
E)  None of these

Question 8

Solve the differential equation using the method of variation of parameters.
$$y ^ { tt } - y ^ { t } = e ^ { 9 x }$$

Accepted Answer

A) 
$y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { x } + \frac { e ^ { 9 x } } { 72 }$ 
B)  $y ( x ) = c _ { 1 } + x e ^ { 9 x }$ 
C)  $y ( x ) = c _ { 1 } + ( 1 + x ) x e ^ { 9 x }$ 
D)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { x } + \frac { e ^ { 9 x } } { 72 }$ 
E)  $y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { 9 x }$ 
A) 
$y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { x } + \frac { e ^ { 9 x } } { 72 }$ 
B)  $y ( x ) = c _ { 1 } + x e ^ { 9 x }$ 
C)  $y ( x ) = c _ { 1 } + ( 1 + x ) x e ^ { 9 x }$ 
D)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { x } + \frac { e ^ { 9 x } } { 72 }$ 
E)  $y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { 9 x }$

Question 9

Solve the initial-value problem.
$$y ^ { tt } + 3 y ^ {t } - 4 y = 0 , y ( 0 ) = 2 , y ^ {t } ( 0 ) = 1$$

Accepted Answer

The answer of Solve the initial-value problem.
\[y ^ { tt...

Question 10

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$.

Accepted Answer

A) 
  
B) 
  
C) 
  
A) 
  
B) 
  
C)

Question 11

A spring with a mass of $9 \mathrm {~kg}$ has damping constant 28 and spring constant 192 . Find the damping constant that would produce critical damping.

Accepted Answer

A)  $c = 288 \sqrt { 3 }$ 
B)  $c = 48 \sqrt { 3 }$ 
C)  $c = 2304$ 
D)  $c = 9 \sqrt { 192 }$ 
E)  $c = 192 \sqrt { 9 }$ 
A)  $c = 288 \sqrt { 3 }$ 
B)  $c = 48 \sqrt { 3 }$ 
C)  $c = 2304$ 
D)  $c = 9 \sqrt { 192 }$ 
E)  $c = 192 \sqrt { 9 }$

Question 12

Graph the particular solution and several other solutions.
$$2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$$

Accepted Answer

A) 
  
B) 
  
C) 
  
A) 
  
B) 
  
C)

Question 13

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } + 5 y ^ { t } + 6 y = x ^ { 2 }$$

Accepted Answer

The answer of Solve the differential equation using the method...

Question 14

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } + 9 y = 7 e ^ { 2 x }$$

Accepted Answer

The answer of Solve the differential equation using the method...

Question 15

Solve the differential equation.
$$\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$$

Accepted Answer

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

Question 16

Solve the initial-value problem using the method of undetermined coefficients.
$$y ^ { tt } + y ^ { t } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { t } ( 0 ) = 0$$

Accepted Answer

The answer of Solve the initial-value problem using the method...

Question 17

Solve the differential equation using the method of variation of parameters.
$$y ^ { tt } - 9 y ^ { t } = \frac { 1 } { x }$$

Accepted Answer

The answer of Solve the differential equation using the method...

Question 18

Use power series to solve the differential equation.
$$y ^ {tt } = 4 y$$

Accepted Answer

The answer of Use power series to solve the differential...

Question 19

Solve the differential equation.
$$9 y ^ { tt } + y = 0$$

Accepted Answer

The answer of Solve the differential equation.
\[9 y ^ {...

Question 20

A series circuit consists of a resistor $R = 16 \Omega$, an inductor with $L = 2 H$, a capacitor with $C = 0.006250 F$, and a $12 - \mathrm { V }$ battery. If the initial charge is $0.0008 \mathrm { C }$ and the initial current is 0 , find the current $I ( t )$ at time $t$.

Accepted Answer

A) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } ( 0.006250 \cos ( 4 t ) + 8 \sin ( 4 t ) ) - \frac { 1 } { 2 }$ 
B)  
$I ( t ) = 0.742 e ^ { - 4 t } \sin ( 8 t )$ 
C) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t ) - \frac { 1 } { 2 }$ 
D) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t )$ 
E)  
$I ( t ) = 0.742 e ^ { - 8 t }$ 
A) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } ( 0.006250 \cos ( 4 t ) + 8 \sin ( 4 t ) ) - \frac { 1 } { 2 }$ 
B)  
$I ( t ) = 0.742 e ^ { - 4 t } \sin ( 8 t )$ 
C) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t ) - \frac { 1 } { 2 }$ 
D) 
$I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t )$ 
E)  
$I ( t ) = 0.742 e ^ { - 8 t }$

Solve the differential equation. $8 y ^ { t t } + y ^ { t } = 0$

Solve the differential equation. $y ^ { tt } + 10 y ^ { t } + 41 y = 0$

Solve the initial-value problem. Select the correct answer. $y ^ { tt} - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 8$

A series circuit consists of a resistor $R = 32 \Omega$ , an inductor with $L = 4 H$ , a capacitor with $C = 0.003125 \mathrm {~F}$ , and a $24 - \mathrm { V }$ battery. If the initial charge is $0.0008 \mathrm { C }$ and the initial current is 0 , find the current $I ( t )$ at time $t$ .

Solve the differential equation. $y ^ { tt } + 4 y ^ { t} - 5 y = 0$

Solve the initial-value problem. $y ^ {tt } + 3 y ^ { t } - 4 y = 0 , y ( 0 ) = 2 , y ^ { t } ( 0 ) = 1$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - y ^ { t } = e ^ { 9 x }$

Solve the initial-value problem. $y ^ { tt } + 3 y ^ {t } - 4 y = 0 , y ( 0 ) = 2 , y ^ {t } ( 0 ) = 1$

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$ .

A spring with a mass of $9 \mathrm {~kg}$ has damping constant 28 and spring constant 192 . Find the damping constant that would produce critical damping.

Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 5 y ^ { t } + 6 y = x ^ { 2 }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 9 y = 7 e ^ { 2 x }$

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { tt } + y ^ { t } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { t } ( 0 ) = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 9 y ^ { t } = \frac { 1 } { x }$

Use power series to solve the differential equation. $y ^ {tt } = 4 y$

Solve the differential equation. $9 y ^ { tt } + y = 0$

A series circuit consists of a resistor $R = 16 \Omega$ , an inductor with $L = 2 H$ , a capacitor with $C = 0.006250 F$ , and a $12 - \mathrm { V }$ battery. If the initial charge is $0.0008 \mathrm { C }$ and the initial current is 0 , find the current $I ( t )$ at time $t$ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 17: Second-Order Differential Equations

Solve the differential equation. 8ytt+yt=08 y ^ { t t } + y ^ { t } = 08ytt+yt=0

Solve the differential equation. ytt+10yt+41y=0y ^ { tt } + 10 y ^ { t } + 41 y = 0ytt+10yt+41y=0

Solve the initial-value problem. Select the correct answer. ytt−2yt−24y=0,y(1)=4,yt(1)=8y ^ { tt} - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 8ytt−2yt−24y=0,y(1)=4,yt(1)=8

Solve the differential equation. ytt+4yt−5y=0y ^ { tt } + 4 y ^ { t} - 5 y = 0ytt+4yt−5y=0

Solve the initial-value problem. ytt+3yt−4y=0,y(0)=2,yt(0)=1y ^ {tt } + 3 y ^ { t } - 4 y = 0 , y ( 0 ) = 2 , y ^ { t } ( 0 ) = 1ytt+3yt−4y=0,y(0)=2,yt(0)=1

Solve the differential equation using the method of variation of parameters. ytt−yt=e9xy ^ { tt } - y ^ { t } = e ^ { 9 x }ytt−yt=e9x

Solve the initial-value problem. ytt+3yt−4y=0,y(0)=2,yt(0)=1y ^ { tt } + 3 y ^ {t } - 4 y = 0 , y ( 0 ) = 2 , y ^ {t } ( 0 ) = 1ytt+3yt−4y=0,y(0)=2,yt(0)=1

A spring with a mass of 2 kg2 \mathrm {~kg}2 kg has damping constant 8 and spring constant 80 . Graph the position function of the mass at time ttt if it starts at the equilibrium position with a velocity of 2 m/s2 \mathrm {~m} / \mathrm { s }2 m/s .

A spring with a mass of 9 kg9 \mathrm {~kg}9 kg has damping constant 28 and spring constant 192 . Find the damping constant that would produce critical damping.

Graph the particular solution and several other solutions. 2y′′+3y′+y=2+cos⁡2x2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x2y′′+3y′+y=2+cos2x

Solve the differential equation using the method of undetermined coefficients. ytt+5yt+6y=x2y ^ { tt } + 5 y ^ { t } + 6 y = x ^ { 2 }ytt+5yt+6y=x2

Solve the differential equation using the method of undetermined coefficients. ytt+9y=7e2xy ^ { tt } + 9 y = 7 e ^ { 2 x }ytt+9y=7e2x

Solve the differential equation. d2ydt2+dydt+4y=0\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0dt2d2y​+dtdy​+4y=0

Solve the initial-value problem using the method of undetermined coefficients. ytt+yt−2y=x+sin⁡x,y(0)=1920,yt(0)=0y ^ { tt } + y ^ { t } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { t } ( 0 ) = 0ytt+yt−2y=x+sinx,y(0)=2019​,yt(0)=0

Solve the differential equation using the method of variation of parameters. ytt−9yt=1xy ^ { tt } - 9 y ^ { t } = \frac { 1 } { x }ytt−9yt=x1​

Use power series to solve the differential equation. ytt=4yy ^ {tt } = 4 yytt=4y

Solve the differential equation. 9ytt+y=09 y ^ { tt } + y = 09ytt+y=0

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Solve the differential equation. $8 y ^ { t t } + y ^ { t } = 0$

Solve the differential equation. $y ^ { tt } + 10 y ^ { t } + 41 y = 0$

Solve the initial-value problem. Select the correct answer. $y ^ { tt} - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 8$

Solve the differential equation. $y ^ { tt } + 4 y ^ { t} - 5 y = 0$

Solve the initial-value problem. $y ^ {tt } + 3 y ^ { t } - 4 y = 0 , y ( 0 ) = 2 , y ^ { t } ( 0 ) = 1$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - y ^ { t } = e ^ { 9 x }$

Solve the initial-value problem. $y ^ { tt } + 3 y ^ {t } - 4 y = 0 , y ( 0 ) = 2 , y ^ {t } ( 0 ) = 1$

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$ .

A spring with a mass of $9 \mathrm {~kg}$ has damping constant 28 and spring constant 192 . Find the damping constant that would produce critical damping.

Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 5 y ^ { t } + 6 y = x ^ { 2 }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 9 y = 7 e ^ { 2 x }$

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { tt } + y ^ { t } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { t } ( 0 ) = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 9 y ^ { t } = \frac { 1 } { x }$

Use power series to solve the differential equation. $y ^ {tt } = 4 y$

Solve the differential equation. $9 y ^ { tt } + y = 0$