Question 1

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

Accepted Answer

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

Question 2

Solve the initial-value problem. $y ^ { tt } + 8 y ^ { t } + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 5$

Accepted Answer

A) 
$y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 9 } { 5 } \sin ( 5 x ) \right)$ 
B) 
$y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
C) 
$y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 9 } { 5 } \sin ( 5 x ) \right)$ 
D) 
$y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
E) 
$y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 9 } { 5 } \sin ( 4 x ) \right)$ 
A) 
$y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 9 } { 5 } \sin ( 5 x ) \right)$ 
B) 
$y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
C) 
$y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 9 } { 5 } \sin ( 5 x ) \right)$ 
D) 
$y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
E) 
$y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 9 } { 5 } \sin ( 4 x ) \right)$

Question 3

Solve the differential equation. $$y ^ { tt } - 4 y ^ { t } + 20 y = 0$$

Accepted Answer

A)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 4 x ) + c _ { 2 } \sin ( 4 x ) \right)$ 
B)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } x \sin ( 2 x ) \right)$ 
C)  $y ( x ) = e ^ { 4 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } \sin ( 2 x ) \right)$ 
D)  $y ( x ) = \operatorname { ce } 2 x \cos ( 4 x )$ 
E)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 4 x ) + c _ { 2 } x \sin ( 4 x ) \right)$ 
A)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 4 x ) + c _ { 2 } \sin ( 4 x ) \right)$ 
B)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } x \sin ( 2 x ) \right)$ 
C)  $y ( x ) = e ^ { 4 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } \sin ( 2 x ) \right)$ 
D)  $y ( x ) = \operatorname { ce } 2 x \cos ( 4 x )$ 
E)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 4 x ) + c _ { 2 } x \sin ( 4 x ) \right)$

Question 4

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

Accepted Answer

A) $y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$ 
B) $y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 15 t } { 2 } } \right) \right]$ 
C) $y ( t ) = e ^ { \frac { \sqrt { t } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$ 
D) $y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right)$ 
E) $y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$ 
A) $y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$ 
B) $y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 15 t } { 2 } } \right) \right]$ 
C) $y ( t ) = e ^ { \frac { \sqrt { t } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$ 
D) $y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right)$ 
E) $y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$

Question 5

Solve the differential equation. $$9 y ^ { tt } = 2 y ^ { t }$$

Accepted Answer

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

Question 6

A spring with a $3 - \mathrm { kg }$ mass is held stretched $0.9 \mathrm {~m}$ beyond its natural length by a force of $30 \mathrm {~N}$. If the spring begins at its equilibrium position but a push gives it an initial velocity of $1 \mathrm {~m} / \mathrm { s }$, find the position $x ( t )$ of the mass after $t$ seconds.

Accepted Answer

A)  $x ( t ) = 1 \sin ( 7 t )$ 
B)  $x ( t ) = 0.3 \sin \left( \frac { 10 } { 3 } t \right) + 0.4 \cos \left( \frac { 10 } { 3 } t \right)$ 
C) $x ( t ) = 0.3 \sin \left( \frac { 10 } { 3 } t \right)$ 
D) $x ( t ) = 0.4 \cos \left( \frac { 10 } { 3 } t \right)$ 
E)  $x ( t ) = \frac { 10 } { 3 } \sin ( 7 t )$ 
A)  $x ( t ) = 1 \sin ( 7 t )$ 
B)  $x ( t ) = 0.3 \sin \left( \frac { 10 } { 3 } t \right) + 0.4 \cos \left( \frac { 10 } { 3 } t \right)$ 
C) $x ( t ) = 0.3 \sin \left( \frac { 10 } { 3 } t \right)$ 
D) $x ( t ) = 0.4 \cos \left( \frac { 10 } { 3 } t \right)$ 
E)  $x ( t ) = \frac { 10 } { 3 } \sin ( 7 t )$

Question 7

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 8

A series circuit consists of a resistor $R = 24 \Omega$, an inductor with $L = 2 H$, a capacitor with $C = 0.005 \mathrm {~F}$, and a generator producing a voltage of $E ( t ) = 12 \cos ( 10 t )$. If the initial charge is $Q = 0.001 \mathrm { C }$ and the initial current is 0 , find the charge $Q ( t )$ at time $t$.

Accepted Answer

A)  $Q ( t ) = e ^ { - 6 t } ( \cos ( 6 t ) - 0.06175 \sin ( 6 t ) )$ 
B) $Q ( t ) = e ^ { - 2 t } ( 0.051 \cos ( 2 t ) - 0.06175 \sin ( 2 t ) ) + \frac { 1 } { 20 } \sin ( 10 t )$ 
C) $Q ( t ) = e ^ { - 6 t } ( 0.001 \cos ( 8 t ) - 0.06175 \sin ( 8 t ) ) + \frac { 1 } { 20 } \sin ( 10 t )$ 
D)  $Q ( t ) = e ^ { - 6 t } ( 0.001 \cos ( 8 t ) - 0.06175 \sin ( 8 t ) )$ 
E)  $Q ( t ) = e ^ { - 3 t } ( \cos ( 5 t ) - 0.06175 \sin ( 5 t ) )$ 
A)  $Q ( t ) = e ^ { - 6 t } ( \cos ( 6 t ) - 0.06175 \sin ( 6 t ) )$ 
B) $Q ( t ) = e ^ { - 2 t } ( 0.051 \cos ( 2 t ) - 0.06175 \sin ( 2 t ) ) + \frac { 1 } { 20 } \sin ( 10 t )$ 
C) $Q ( t ) = e ^ { - 6 t } ( 0.001 \cos ( 8 t ) - 0.06175 \sin ( 8 t ) ) + \frac { 1 } { 20 } \sin ( 10 t )$ 
D)  $Q ( t ) = e ^ { - 6 t } ( 0.001 \cos ( 8 t ) - 0.06175 \sin ( 8 t ) )$ 
E)  $Q ( t ) = e ^ { - 3 t } ( \cos ( 5 t ) - 0.06175 \sin ( 5 t ) )$

Question 9

The solution of the initial-value problem $x ^ { 2 } y ^ { tt } + x y ^ { t } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.

Accepted Answer

A) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n x ^ { 2 n - 1 } } { 2 ^ { 2 n - 1 } ( n ! ) ^ { 2 } }$ 
B) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { n } } { 2 ^ { n } ( 2 n ! ) }$ 
C) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$ 
D) $y ( x ) = 2 ^ { x } + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( n ! ) ^ { 2 } }$ 
E) $y ( x ) = x + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$ 
A) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n x ^ { 2 n - 1 } } { 2 ^ { 2 n - 1 } ( n ! ) ^ { 2 } }$ 
B) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { n } } { 2 ^ { n } ( 2 n ! ) }$ 
C) $y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$ 
D) $y ( x ) = 2 ^ { x } + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( n ! ) ^ { 2 } }$ 
E) $y ( x ) = x + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$

Question 10

Solve the initial-value problem. $$2 y ^ { tt} + 21 y ^ { t } + 54 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 1$$

Accepted Answer

The answer of Solve the initial-value problem. \[2 y ^...

Question 11

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

Accepted Answer

A) $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 9 } { 108 } c _ { 3 }$ 
B) $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 1 } { 6 } x ^ { 2 }$ 
C)  $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x }$ 
D)  $y ( x ) = \frac { 1 } { 6 } x ^ { 2 } - \frac { 5 } { 18 } x + \frac { 19 } { 108 }$ 
E)  $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 1 } { 3 } x ^ { 2 } - \frac { 5 } { 9 } x + \frac { 19 } { 54 }$ 
A) $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 9 } { 108 } c _ { 3 }$ 
B) $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 1 } { 6 } x ^ { 2 }$ 
C)  $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x }$ 
D)  $y ( x ) = \frac { 1 } { 6 } x ^ { 2 } - \frac { 5 } { 18 } x + \frac { 19 } { 108 }$ 
E)  $y ( x ) = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x } + \frac { 1 } { 3 } x ^ { 2 } - \frac { 5 } { 9 } x + \frac { 19 } { 54 }$

Question 12

Solve the differential equation using the method of variation of parameters. $$y ^ { tt } - 4 y ^ {t } + 3 y = 2 \sin x$$

Accepted Answer

A)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 1 } { 5 } \sin x$ 
B)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 2 } { 5 } \cos x + \frac { 1 } { 5 } \sin x$ 
C)  $y ( x ) = c _ { 1 } \sin x + \frac { c _ { 2 } } { 5 } x + \frac { 1 } { 5 } \sin x$ 
D)  $y ( x ) = c _ { 1 } \sin 3 x + \frac { c _ { 2 } } { 4 } x + \cos 3 x$ 
E)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 2 } { 5 } \sin x$ 
A)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 1 } { 5 } \sin x$ 
B)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 2 } { 5 } \cos x + \frac { 1 } { 5 } \sin x$ 
C)  $y ( x ) = c _ { 1 } \sin x + \frac { c _ { 2 } } { 5 } x + \frac { 1 } { 5 } \sin x$ 
D)  $y ( x ) = c _ { 1 } \sin 3 x + \frac { c _ { 2 } } { 4 } x + \cos 3 x$ 
E)  $y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 2 } { 5 } \sin x$

Question 13

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 $12 - V$ 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)  $I ( t ) = 0.742 e ^ { - 8 t }$

Question 14

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 15

Solve the initial-value problem. $$y ^ { tt } + 16 y = 0 , y \left( \frac { \pi } { 4 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 4 } \right) = 1$$

Accepted Answer

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

Question 16

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

Accepted Answer

A) 
  
B) 
  
C)

Question 17

Solve the boundary-value problem, if possible. $$y ^ { tt } - y ^ {t } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$$

Accepted Answer

The answer of Solve the boundary-value problem, if possible. \[y...

Question 18

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 19

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 20

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

Accepted Answer

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

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

Solve the initial-value problem. $y ^ { tt } + 8 y ^ { t } + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 5$

Solve the differential equation. $y ^ { tt } - 4 y ^ { t } + 20 y = 0$

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

Solve the differential equation. $9 y ^ { tt } = 2 y ^ { t }$

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

The solution of the initial-value problem $x ^ { 2 } y ^ { tt } + x y ^ { t } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.

Solve the initial-value problem. $2 y ^ { tt} + 21 y ^ { t } + 54 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 1$

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

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 4 y ^ {t } + 3 y = 2 \sin x$

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 $12 - V$ 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 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 initial-value problem. $y ^ { tt } + 16 y = 0 , y \left( \frac { \pi } { 4 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 4 } \right) = 1$

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

Solve the boundary-value problem, if possible. $y ^ { tt } - y ^ {t } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$

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

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

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

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 ^ { tt } + y ^ { t } = 08ytt+yt=0

Solve the initial-value problem. ytt+8yt+41y=0,y(0)=1,yt(0)=5y ^ { tt } + 8 y ^ { t } + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 5ytt+8yt+41y=0,y(0)=1,yt(0)=5

Solve the differential equation. ytt−4yt+20y=0y ^ { tt } - 4 y ^ { t } + 20 y = 0ytt−4yt+20y=0

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 differential equation. 9ytt=2yt9 y ^ { tt } = 2 y ^ { t }9ytt=2yt

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

Solve the initial-value problem. 2ytt+21yt+54y=0,y(0)=0,yt(0)=12 y ^ { tt} + 21 y ^ { t } + 54 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 12ytt+21yt+54y=0,y(0)=0,yt(0)=1

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

Solve the differential equation using the method of variation of parameters. ytt−4yt+3y=2sin⁡xy ^ { tt } - 4 y ^ {t } + 3 y = 2 \sin xytt−4yt+3y=2sinx

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 initial-value problem. ytt+16y=0,y(π4)=0,yt(π4)=1y ^ { tt } + 16 y = 0 , y \left( \frac { \pi } { 4 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 4 } \right) = 1ytt+16y=0,y(4π​)=0,yt(4π​)=1

Graph the particular solution and several other solutions. 2ytt+3yt+y=2+cos⁡2x2 y ^ { tt } + 3 y ^ { t } + y = 2 + \cos 2 x2ytt+3yt+y=2+cos2x

Solve the boundary-value problem, if possible. ytt−yt−2y=0,y(−1)=1,y(1)=0y ^ { tt } - y ^ {t } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0ytt−yt−2y=0,y(−1)=1,y(1)=0

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 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 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

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. $8 y ^ { tt } + y ^ { t } = 0$

Solve the initial-value problem. $y ^ { tt } + 8 y ^ { t } + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 5$

Solve the differential equation. $y ^ { tt } - 4 y ^ { t } + 20 y = 0$

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

Solve the differential equation. $9 y ^ { tt } = 2 y ^ { t }$

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

Solve the initial-value problem. $2 y ^ { tt} + 21 y ^ { t } + 54 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 1$

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

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 4 y ^ {t } + 3 y = 2 \sin x$

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 initial-value problem. $y ^ { tt } + 16 y = 0 , y \left( \frac { \pi } { 4 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 4 } \right) = 1$

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

Solve the boundary-value problem, if possible. $y ^ { tt } - y ^ {t } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$

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

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

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