Question 1

Solve the initial-value problem. $$2 y ^ { tt } + 19 y ^ {t } + 24 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 8$$

Accepted Answer

$$\frac { 16 } { 13 } e ^ { - \frac { 3 x } { 2 } } - \frac { 16 } { 13 } e ^ { - 3 x }$$

Question 2

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)  $c = 195 \sqrt { 6 }$ C

Question 3

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

Accepted Answer

A) $y ( x ) = c e ^ { - \frac { 4 x ^ { 2 } } { 2 } }$ 
B)  $y ( x ) = c e ^ { \frac { 4 x ^ { 2 } } { 2 } }$ 
C)  $y ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$ 
D)  $y ( x ) = c e ^ { 4 x }$ 
E) $y ( x ) = 4 e ^ { \frac { x ^ { 2 } } { 2 } }$ 
A) $y ( x ) = c e ^ { - \frac { 4 x ^ { 2 } } { 2 } }$ 
B)  $y ( x ) = c e ^ { \frac { 4 x ^ { 2 } } { 2 } }$ 
C)  $y ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$ 
D)  $y ( x ) = c e ^ { 4 x }$ 
E) $y ( x ) = 4 e ^ { \frac { x ^ { 2 } } { 2 } }$ B

Question 4

Solve the initial-value problem using the method of undetermined coefficients. $$y ^ { tt } - y ^ { t } = x e ^ { x } , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 3$$

Accepted Answer

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

Question 5

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)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$ 
E)  None of these 
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)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$ 
E)  None of these

Question 6

Solve the boundary-value problem, if possible. $$y ^ { tt } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3$$

Accepted Answer

A)  $y ( x ) = 7 x e ^ { ( - 3 x + 3 ) }$ 
B)  $y ( x ) = 3 e ^ { ( - 7 x - 7 ) }$ 
C)  $y ( x ) = 3 x e ^ { ( - 7 x - 7 ) }$ 
D)  $y ( x ) = 3 x e ^ { ( - 7 x + 7 ) }$ 
E)  No solution 
A)  $y ( x ) = 7 x e ^ { ( - 3 x + 3 ) }$ 
B)  $y ( x ) = 3 e ^ { ( - 7 x - 7 ) }$ 
C)  $y ( x ) = 3 x e ^ { ( - 7 x - 7 ) }$ 
D)  $y ( x ) = 3 x e ^ { ( - 7 x + 7 ) }$ 
E)  No solution

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

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

Question 8

Solve the differential equation. $$y ^ {tt } - 8 y ^ { t } + 32 y = 0$$

Accepted Answer

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

Question 9

Solve the boundary-value problem, if possible. $$y ^ { tt } + 10 y ^ { t } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 9$$

Accepted Answer

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

Question 10

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 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

The solution of the initial-value problem $x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ 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 14

Solve the boundary-value problem, if possible. $$y ^ {tt } + 5 y ^ {t } - 24 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$$

Accepted Answer

A)  $y ( x ) = \left( e ^ { 6 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
B)  $y ( x ) = \left( e ^ { 8 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { x } - e ^ { - 8 x } \right)$ 
C)  $y ( x ) = \left( e ^ { 3 } - e ^ { - 16 } \right) \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
D)  $y ( x ) = \left( e ^ { 3 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
E)  No solution 
A)  $y ( x ) = \left( e ^ { 6 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
B)  $y ( x ) = \left( e ^ { 8 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { x } - e ^ { - 8 x } \right)$ 
C)  $y ( x ) = \left( e ^ { 3 } - e ^ { - 16 } \right) \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
D)  $y ( x ) = \left( e ^ { 3 } - e ^ { - 16 } \right) ^ { - 1 } \left( e ^ { 3 x } - e ^ { - 8 x } \right)$ 
E)  No solution

Question 15

Solve the differential equation. $$y ^ { tt } - 2 y ^ { t } = e ^ { 3 x }$$

Accepted Answer

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

Question 16

Solve the differential equation using the method of undetermined coefficients. $$y ^ { tt } - 5 y ^ { t } = \sin 15 x$$

Accepted Answer

A)  $y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 250 } \cos ( 6 x ) + \frac { 1 } { 750 } \sin ( 6 x )$ 
B) $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$ 
C)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } - \frac { 1 } { 250 } \sin ( 6 x )$ 
D) $y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$ 
E) $y ( x ) = \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$ 
A)  $y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 250 } \cos ( 6 x ) + \frac { 1 } { 750 } \sin ( 6 x )$ 
B) $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$ 
C)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } - \frac { 1 } { 250 } \sin ( 6 x )$ 
D) $y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$ 
E) $y ( x ) = \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$

Question 17

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 $F ( t ) = 6 F _ { 0 } \cos ( \omega t )$ 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

A) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } e ^ { - \omega t } } { 2 M \omega }$ 
B) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$ 
C) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 3 F _ { 0 } t } { M \omega } \sin ( \omega t )$ 
D) $x ( t ) = \frac { F _ { 0 } t } { 2 M \omega } \left( c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) \right)$ 
E) $x ( t ) = \frac { F _ { 0 } t ^ { 2 } } { 2 M \omega } \cos ( \omega t )$ 
A) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } e ^ { - \omega t } } { 2 M \omega }$ 
B) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$ 
C) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 3 F _ { 0 } t } { M \omega } \sin ( \omega t )$ 
D) $x ( t ) = \frac { F _ { 0 } t } { 2 M \omega } \left( c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) \right)$ 
E) $x ( t ) = \frac { F _ { 0 } t ^ { 2 } } { 2 M \omega } \cos ( \omega t )$

Question 18

Solve the initial-value problem. $$y ^ { tt } + 8 y ^ {t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t} ( 0 ) = 2$$

Accepted Answer

A) $y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 6 } { 5 } \sin ( 5 x ) \right)$ 
B) $y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 6 } { 5 } \sin ( 5 x ) \right)$ 
C) $y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 6 } { 5 } \sin ( 4 x ) \right)$ 
D) $y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
E) $y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
A) $y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 6 } { 5 } \sin ( 5 x ) \right)$ 
B) $y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 6 } { 5 } \sin ( 5 x ) \right)$ 
C) $y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 6 } { 5 } \sin ( 4 x ) \right)$ 
D) $y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$ 
E) $y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$

Question 19

Use power series to solve the differential equation. $$y ^ {tt } = 36 y$$

Accepted Answer

A)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x$ 
B)  $y ( x ) = 6 \cosh x + 6 \sinh x$ 
C)  $y ( x ) = c _ { 1 } 6 \cosh x + c _ { 2 } 6 \sinh x + e ^ { x }$ 
D)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x + e ^ { x }$ 
E)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x + x ^ { 2 }$ 
A)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x$ 
B)  $y ( x ) = 6 \cosh x + 6 \sinh x$ 
C)  $y ( x ) = c _ { 1 } 6 \cosh x + c _ { 2 } 6 \sinh x + e ^ { x }$ 
D)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x + e ^ { x }$ 
E)  $y ( x ) = c _ { 1 } \cosh 6 x + c _ { 2 } \sinh 6 x + x ^ { 2 }$

Question 20

Solve the differential equation using the method of undetermined coefficients. $$y ^ { tt} - 3 y ^ { t } = \sin 9 x$$

Accepted Answer

A)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 3 x } - \frac { 1 } { 90 } \sin ( 6 x )$ 
B)  $y ( x ) = c _ { 1 } e ^ { 3 x } + \frac { 1 } { 90 } \cos ( 6 x ) + \frac { 1 } { 270 } \sin ( 6 x )$ 
C)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 3 x } + \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$ 
D)  $y ( x ) = c _ { 1 } e ^ { 3 x } + \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$ 
E)  $y ( x ) = \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$ 
A)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 3 x } - \frac { 1 } { 90 } \sin ( 6 x )$ 
B)  $y ( x ) = c _ { 1 } e ^ { 3 x } + \frac { 1 } { 90 } \cos ( 6 x ) + \frac { 1 } { 270 } \sin ( 6 x )$ 
C)  $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 3 x } + \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$ 
D)  $y ( x ) = c _ { 1 } e ^ { 3 x } + \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$ 
E)  $y ( x ) = \frac { 1 } { 270 } \cos ( 6 x ) - \frac { 1 } { 90 } \sin ( 6 x )$

Solve the initial-value problem. $2 y ^ { tt } + 19 y ^ {t } + 24 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 8$

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.

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

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { tt } - y ^ { t } = x e ^ { x } , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 3$

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

Solve the boundary-value problem, if possible. $y ^ { tt } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3$

Solve the differential equation. $y ^ {tt } - 8 y ^ { t } + 32 y = 0$

Solve the boundary-value problem, if possible. $y ^ { tt } + 10 y ^ { t } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 9$

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

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

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

The solution of the initial-value problem $x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ 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 boundary-value problem, if possible. $y ^ {tt } + 5 y ^ {t } - 24 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

Solve the differential equation. $y ^ { tt } - 2 y ^ { t } = e ^ { 3 x }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 5 y ^ { t } = \sin 15 x$

Solve the initial-value problem. $y ^ { tt } + 8 y ^ {t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t} ( 0 ) = 2$

Use power series to solve the differential equation. $y ^ {tt } = 36 y$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt} - 3 y ^ { t } = \sin 9 x$

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Exam 17: Second-Order Differential Equations

Solve the initial-value problem. 2ytt+19yt+24y=0,y(0)=0,yt(0)=82 y ^ { tt } + 19 y ^ {t } + 24 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 82ytt+19yt+24y=0,y(0)=0,yt(0)=8

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.

Use power series to solve the differential equation. yt=4xy.y ^ { t } = 4 x y .yt=4xy.

Solve the initial-value problem using the method of undetermined coefficients. ytt−yt=xex,y(0)=0,yt(0)=3y ^ { tt } - y ^ { t } = x e ^ { x } , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 3ytt−yt=xex,y(0)=0,yt(0)=3

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

Solve the boundary-value problem, if possible. ytt+14yt+49y=0,y(0)=0,y(1)=3y ^ { tt } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3ytt+14yt+49y=0,y(0)=0,y(1)=3

Solve the differential equation. ytt−8yt+32y=0y ^ {tt } - 8 y ^ { t } + 32 y = 0ytt−8yt+32y=0

Solve the boundary-value problem, if possible. ytt+10yt+25y=0,y(0)=0,y(1)=9y ^ { tt } + 10 y ^ { t } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 9ytt+10yt+25y=0,y(0)=0,y(1)=9

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

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+5yt−24y=0,y(0)=0,y(2)=1y ^ {tt } + 5 y ^ {t } - 24 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1ytt+5yt−24y=0,y(0)=0,y(2)=1

Solve the differential equation. ytt−2yt=e3xy ^ { tt } - 2 y ^ { t } = e ^ { 3 x }ytt−2yt=e3x

Solve the differential equation using the method of undetermined coefficients. ytt−5yt=sin⁡15xy ^ { tt } - 5 y ^ { t } = \sin 15 xytt−5yt=sin15x

Solve the initial-value problem. ytt+8yt+41y=0,y(0)=1,yt(0)=2y ^ { tt } + 8 y ^ {t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t} ( 0 ) = 2ytt+8yt+41y=0,y(0)=1,yt(0)=2

Use power series to solve the differential equation. ytt=36yy ^ {tt } = 36 yytt=36y

Solve the differential equation using the method of undetermined coefficients. ytt−3yt=sin⁡9xy ^ { tt} - 3 y ^ { t } = \sin 9 xytt−3yt=sin9x

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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

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Solve the initial-value problem. $2 y ^ { tt } + 19 y ^ {t } + 24 y = 0 , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 8$

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.

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

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { tt } - y ^ { t } = x e ^ { x } , y ( 0 ) = 0 , y ^ { t } ( 0 ) = 3$

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

Solve the boundary-value problem, if possible. $y ^ { tt } + 14 y ^ { t } + 49 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 3$

Solve the differential equation. $y ^ {tt } - 8 y ^ { t } + 32 y = 0$

Solve the boundary-value problem, if possible. $y ^ { tt } + 10 y ^ { t } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 9$

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

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

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. $y ^ {tt } + 5 y ^ {t } - 24 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

Solve the differential equation. $y ^ { tt } - 2 y ^ { t } = e ^ { 3 x }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 5 y ^ { t } = \sin 15 x$

Solve the initial-value problem. $y ^ { tt } + 8 y ^ {t} + 41 y = 0 , y ( 0 ) = 1 , y ^ { t} ( 0 ) = 2$

Use power series to solve the differential equation. $y ^ {tt } = 36 y$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt} - 3 y ^ { t } = \sin 9 x$