Question 1

Use power series to solve the differential equation. Select the correct answer.
$$\left( x ^ { 2 } + 1 \right) y ^ { tt } + x y ^ { t } - y = 0$$

Accepted Answer

A) 
$y ( x ) = c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$ 
B) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$ 
C) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } ( n - 2 ) ! } x ^ { 2 n }$ 
D) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$ 
E) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 2 n ) ! } { 2 ^ { 2 n + 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$ 
A) 
$y ( x ) = c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$ 
B) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$ 
C) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } ( n - 2 ) ! } x ^ { 2 n }$ 
D) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$ 
E) 
$y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n - 2 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 2 n ) ! } { 2 ^ { 2 n + 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$

Question 2

Solve the initial-value problem. Select the correct answer.
$$y ^ { tt } + 8 y ^ { t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 4$$

Accepted Answer

A) 
$$y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 8 } { 5 } \sin ( 5 x ) \right)$$ 
B) 
$$y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$$ 
C) 
$$y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 8 } { 5 } \sin ( 4 x ) \right)$$ 
D) 
$$y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$$ 
E) 
$$y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 8 } { 5 } \sin ( 5 x ) \right)$$ 
A) 
$$y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 8 } { 5 } \sin ( 5 x ) \right)$$ 
B) 
$$y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$$ 
C) 
$$y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 8 } { 5 } \sin ( 4 x ) \right)$$ 
D) 
$$y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$$ 
E) 
$$y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 8 } { 5 } \sin ( 5 x ) \right)$$

Question 3

Solve the differential equation.
$$y ^ { tt } - 4 y ^ { t } + 13 y = 0$$

Accepted Answer

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

Question 4

A spring with a mass of $6 \mathrm {~kg}$ has damping constant 28 and spring constant 195 . Find the damping constant that would produce critical damping.

Accepted Answer

A)  $c = 6 \sqrt { 195 }$ 
B)  $c = 6 \sqrt { 130 }$ 
C)  $c = 780 \sqrt { 6 }$ 
D)  $c = 24 \sqrt { 195 }$ 
E)  $c = 195 \sqrt { 6 }$ 
A)  $c = 6 \sqrt { 195 }$ 
B)  $c = 6 \sqrt { 130 }$ 
C)  $c = 780 \sqrt { 6 }$ 
D)  $c = 24 \sqrt { 195 }$ 
E)  $c = 195 \sqrt { 6 }$

Question 5

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

Accepted Answer

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

Question 6

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } + 6 y ^ { t } + 9 y = 2 + x$$

Accepted Answer

A) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x$ 
B)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x }$ 
C) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$ 
D) 
$y ( x ) = c _ { 1 } x e ^ { - 3 x } + c _ { 2 } x ^ { 2 } e ^ { - 3 x } + \frac { 4 } { 27 } x$ 
E) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$ 
A) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x$ 
B)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x }$ 
C) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$ 
D) 
$y ( x ) = c _ { 1 } x e ^ { - 3 x } + c _ { 2 } x ^ { 2 } e ^ { - 3 x } + \frac { 4 } { 27 } x$ 
E) 
$y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$

Question 7

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 $t$.

Accepted Answer

The answer of A spring with a \(16 - \mathrm...

Question 8

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 9

Use power series to solve the differential equation.
$$y ^ { tt } - x y ^ { t} - y = 0 , y ( 0 ) = 6 , y ^ {t } ( 0 ) = 0$$

Accepted Answer

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

Question 10

Solve the differential equation. Select the correct answer.
$$y ^ { tt } - 8 y ^ { t } + 25 y = 0$$

Accepted Answer

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

Question 11

A spring with a 16-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 $t$.

Accepted Answer

The answer of A spring with a 16-kg mass has...

Question 12

Solve the differential equation using the method of variation of parameters.
$$y ^ { tt } + 25 y = x$$

Accepted Answer

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

Question 13

Solve the boundary-value problem, if possible.
$$y ^ { tt } + 16 y ^ { t} + 64 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 5$$

Accepted Answer

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

Question 14

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients.
$$y ^ {tt } + 3 y ^ { t } + 5 y = x ^ { 4 } e ^ { 9 x }$$

Accepted Answer

The answer of Find a trial solution for the method...

Question 15

Use power series to solve the differential equation. Select the correct answer.
$$y ^ { tt } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { t } ( 0 ) = 0$$

Accepted Answer

A) 
$$y ( x ) = 6 - \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
B) 
$$y ( x ) = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 6 x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
C) 
$$y ( x ) = 6 + \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 6 x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
D) 
$$y ( x ) = - 6 + \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
E) 
$$y ( x ) = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
A) 
$$y ( x ) = 6 - \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
B) 
$$y ( x ) = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 6 x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
C) 
$$y ( x ) = 6 + \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 6 x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
D) 
$$y ( x ) = - 6 + \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$ 
E) 
$$y ( x ) = \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 4 n } } { 4 n ( 4 n - 1 ) \cdot ( 4 n - 4 ) ( 4 n - 5 ) \cdots 4 \cdot 3 }$$

Question 16

Solve the initial-value problem.
$$y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$$

Accepted Answer

A)  $y ( x ) = \frac { 22 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 18 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
B)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
C) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
D) 
$y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$ 
E) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$ 
A)  $y ( x ) = \frac { 22 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 18 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
B)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
C) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
D) 
$y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$ 
E) 
$y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$

Question 17

Solve the initial-value problem. Select the correct answer.
$$y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$$

Accepted Answer

A)  $y ( x ) = \frac { 22 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 18 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
B)  $y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
C)  $y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$ 
D)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
E)  $y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$ 
A)  $y ( x ) = \frac { 22 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 18 } { 10 } e ^ { - 4 ( x - 1 ) }$ 
B)  $y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$ 
C)  $y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$ 
D)  $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$ 
E)  $y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$

Question 18

Suppose a spring has mass $M$ and spring constant $k$ and let $\omega = \sqrt { k / M }$. Suppose that the damping constant is so small that the damping force is negligible. If an external force $F ( t ) = 2 F _ { 0 } \cos ( \omega t )$ is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to find the equation that describes the motion of the mass.

Accepted Answer

The answer of Suppose a spring has mass $M$ and...

Question 19

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

Accepted Answer

A) 
$y ( x ) = c _ { 1 } \sin 3 x + \frac { c _ { 2 } } { 4 } x + \cos 3 x$ 
B) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 5 } \cos x + \frac { 3 } { 10 } \sin x$ 
C) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 5 } \sin x$ 
D) 
$y ( x ) = c _ { 1 } \sin x + \frac { c _ { 2 } } { 5 } x + \frac { 3 } { 10 } \sin x$ 
E) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 10 } \sin x$ 
A) 
$y ( x ) = c _ { 1 } \sin 3 x + \frac { c _ { 2 } } { 4 } x + \cos 3 x$ 
B) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 5 } \cos x + \frac { 3 } { 10 } \sin x$ 
C) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 5 } \sin x$ 
D) 
$y ( x ) = c _ { 1 } \sin x + \frac { c _ { 2 } } { 5 } x + \frac { 3 } { 10 } \sin x$ 
E) 
$y ( x ) = c _ { 1 } e ^ { 3 x } + c _ { 2 } e ^ { x } + \frac { 3 } { 10 } \sin x$

Question 20

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

Use power series to solve the differential equation. Select the correct answer. $\left( x ^ { 2 } + 1 \right) y ^ { tt } + x y ^ { t } - y = 0$

Solve the initial-value problem. Select the correct answer. $y ^ { tt } + 8 y ^ { t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 4$

Solve the differential equation. $y ^ { tt } - 4 y ^ { t } + 13 y = 0$

A spring with a mass of $6 \mathrm {~kg}$ has damping constant 28 and spring constant 195 . Find the damping constant that would produce critical damping.

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 6 y ^ { t } + 9 y = 2 + x$

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

Use power series to solve the differential equation. $y ^ { tt } - x y ^ { t} - y = 0 , y ( 0 ) = 6 , y ^ {t } ( 0 ) = 0$

Solve the differential equation. Select the correct answer. $y ^ { tt } - 8 y ^ { t } + 25 y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } + 25 y = x$

Solve the boundary-value problem, if possible. $y ^ { tt } + 16 y ^ { t} + 64 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 5$

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ {tt } + 3 y ^ { t } + 5 y = x ^ { 4 } e ^ { 9 x }$

Use power series to solve the differential equation. Select the correct answer. $y ^ { tt } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { t } ( 0 ) = 0$

Solve the initial-value problem. $y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$

Solve the initial-value problem. Select the correct answer. $y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$

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

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

Use power series to solve the differential equation. Select the correct answer. (x2+1)ytt+xyt−y=0\left( x ^ { 2 } + 1 \right) y ^ { tt } + x y ^ { t } - y = 0(x2+1)ytt+xyt−y=0

Solve the initial-value problem. Select the correct answer. ytt+8yt+41y=0,y(0)=1,yt(0)=4y ^ { tt } + 8 y ^ { t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 4ytt+8yt+41y=0,y(0)=1,yt(0)=4

Solve the differential equation. ytt−4yt+13y=0y ^ { tt } - 4 y ^ { t } + 13 y = 0ytt−4yt+13y=0

A spring with a mass of 6 kg6 \mathrm {~kg}6 kg has damping constant 28 and spring constant 195 . Find the damping constant that would produce critical damping.

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

Solve the differential equation using the method of undetermined coefficients. ytt+6yt+9y=2+xy ^ { tt } + 6 y ^ { t } + 9 y = 2 + xytt+6yt+9y=2+x

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

Use power series to solve the differential equation. ytt−xyt−y=0,y(0)=6,yt(0)=0y ^ { tt } - x y ^ { t} - y = 0 , y ( 0 ) = 6 , y ^ {t } ( 0 ) = 0ytt−xyt−y=0,y(0)=6,yt(0)=0

Solve the differential equation. Select the correct answer. ytt−8yt+25y=0y ^ { tt } - 8 y ^ { t } + 25 y = 0ytt−8yt+25y=0

Solve the differential equation using the method of variation of parameters. ytt+25y=xy ^ { tt } + 25 y = xytt+25y=x

Solve the boundary-value problem, if possible. ytt+16yt+64y=0,y(0)=0,y(1)=5y ^ { tt } + 16 y ^ { t} + 64 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 5ytt+16yt+64y=0,y(0)=0,y(1)=5

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. ytt+3yt+5y=x4e9xy ^ {tt } + 3 y ^ { t } + 5 y = x ^ { 4 } e ^ { 9 x }ytt+3yt+5y=x4e9x

Use power series to solve the differential equation. Select the correct answer. ytt+x2y=0,y(0)=6,yt(0)=0y ^ { tt } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { t } ( 0 ) = 0ytt+x2y=0,y(0)=6,yt(0)=0

Solve the initial-value problem. ytt−2yt−24y=0,y(1)=4,yt(1)=6y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6ytt−2yt−24y=0,y(1)=4,yt(1)=6

Solve the initial-value problem. Select the correct answer. ytt−2yt−24y=0,y(1)=4,yt(1)=6y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6ytt−2yt−24y=0,y(1)=4,yt(1)=6

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

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

Use power series to solve the differential equation. Select the correct answer. $\left( x ^ { 2 } + 1 \right) y ^ { tt } + x y ^ { t } - y = 0$

Solve the initial-value problem. Select the correct answer. $y ^ { tt } + 8 y ^ { t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 4$

Solve the differential equation. $y ^ { tt } - 4 y ^ { t } + 13 y = 0$

A spring with a mass of $6 \mathrm {~kg}$ has damping constant 28 and spring constant 195 . Find the damping constant that would produce critical damping.

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 6 y ^ { t } + 9 y = 2 + x$

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

Use power series to solve the differential equation. $y ^ { tt } - x y ^ { t} - y = 0 , y ( 0 ) = 6 , y ^ {t } ( 0 ) = 0$

Solve the differential equation. Select the correct answer. $y ^ { tt } - 8 y ^ { t } + 25 y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } + 25 y = x$

Solve the boundary-value problem, if possible. $y ^ { tt } + 16 y ^ { t} + 64 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 5$

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ {tt } + 3 y ^ { t } + 5 y = x ^ { 4 } e ^ { 9 x }$

Use power series to solve the differential equation. Select the correct answer. $y ^ { tt } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { t } ( 0 ) = 0$

Solve the initial-value problem. $y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$

Solve the initial-value problem. Select the correct answer. $y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$

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