Deck 17: Second-Order Differential Equations

Question 1

A spring with a mass of 2 kg has damping constant 14, and a force of N is required to keep the spring stretched m beyond its natural length. The spring is stretched 1m beyond its natural length and then released with zero velocity. Find the position x(t) of the mass at any time t.

Answer 1

Question 2

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = \frac { e ^ { - 2 x } } { x ^ { 3 } }$

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

Answer 2

$y ( x ) = e ^ { - 2 x } \left( c _ { 1 } + c _ { 2 } x + \frac { 1 } { 2 x } \right)$

Question 3

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

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) $y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 2 n ) ! } { 2 ^ { 2 n + 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$

Answer 3

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

Question 4

Use power series to solve the differential equation. $y ^ { \prime \prime } = 25 y$

A) $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x$
B) $y ( x ) = c _ { 1 } 5 \cosh x + c _ { 2 } 5 \sinh x + e ^ { x }$
C) $y ( x ) = 5 \cosh x + 5 \sinh x$
D) $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + e ^ { x }$
E) $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + x ^ { 2 }$

Question 5

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 $24$ -V battery. If the initial charge is 0.0008 C and the initial current is 0, find the current I(t) at time t.

A) $I ( t ) = 0.742 e ^ { - 4 t } \sin ( 8 t )$
B) $I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t )$
C) $I ( t ) = \frac { e ^ { - 4 t } } { 160 } \cos ( 8 t ) - \frac { 1 } { 2 }$
D) $I ( t ) = 0.742 e ^ { - 8 t }$
E) $I ( t ) = \frac { e ^ { - 4 t } } { 160 } ( 0.003125 \cos ( 4 t ) + 8 \sin ( 4 t ) ) - \frac { 1 } { 2 }$

Question 6

Use power series to solve the differential equation. $y ^ { \prime } = 7 x ^ { 3 } y$

A) $y ( x ) = 7 e ^ { \frac { x ^ { 4 } } { 4 } }$
B) $y ( x ) = c e ^ { 7 x }$
C) $y ( x ) = e ^ { - \frac { x ^ { 4 } } { 4 } }$
D) $y ( x ) = c e ^ { - \frac { 7 x ^ { 4 } } { 4 } }$
E) $y ( x ) = c e ^ { \frac { 7 x ^ { 4 } } { 4 } }$

Question 7

A series circuit consists of a resistor an inductor with L = H, a capacitor with
C = F, and a -V battery. If the initial charge and current are both 0, find the charge Q(t) at time t.

Question 8

A spring has a mass of $1$ kg and its damping constant is $c = 10$ . The spring starts from its equilibrium position with a velocity of $1$ m/s. Graph the position function for the spring constant $k = 20$ .

A)
B)
C)

Question 9

Use power series to solve the differential equation. $$( x - 10 ) y ^ { \prime } + 2 y = 0$$

A) $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 10 ^ { n } }$$
B) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n + 1 } { 10 ^ { n } } x ^ { n }$
C) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { ( n + 1 ) } { x ^ { n } }$
D) $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } ( n + 1 ) x ^ { n }$$
E) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n x ^ { n } } { 10 ^ { n } }$

Question 10

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

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

Question 11

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 $F ( t ) = 8 F _ { 0 } \cos ( \omega t )$ 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.

A) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$
B) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } e ^ { - \omega t } } { 2 M \omega }$
C) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 4 F _ { 0 } t } { M \omega } \sin ( \omega t )$
D) $x ( t ) = \frac { F _ { 0 } t ^ { 2 } } { 2 M \omega } \cos ( \omega t )$
E) $x ( t ) = \frac { F _ { 0 } t } { 2 M \omega } \left( c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) \right)$

Question 12

A spring with a mass of 2 kg has damping constant 14, and a force of N is required to keep the spring stretched m beyond its natural length. Find the mass that would produce critical damping.

Question 13

A spring with a mass of $9$ kg has damping constant 28 and spring constant $195$ . Find the damping constant that would produce critical damping.

A) $c =$ $36 \sqrt { 195 }$
B) $c =$ $2340$
C) $c = 195 \sqrt { 9 }$
D) $c =$ 9 $\sqrt { 195 }$
E) $c =$ $6 \sqrt { 195 }$

Question 14

Use power series to solve the differential equation. $y ^ { \prime \prime } - x y ^ { \prime } - y = 0 , y ( 0 ) = 7 , y ^ { \prime } ( 0 ) = 0$

A) $y ( x ) = 7 e ^ { x }$
B) $y ( x ) = 7 e ^ { - \frac { x ^ { 2 } } { 2 } }$
C) $y ( x ) = e ^ { 7 x }$
D) $y ( x ) = c e ^ { \frac { x ^ { 2 } } { 2 } }$
E) $y ( x ) = 7 e ^ { \frac { x ^ { 2 } } { 2 } }$

Question 15

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

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 16

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

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 17

A series circuit consists of a resistor $R = 96 \Omega$ , an inductor with $L = 8 \mathrm { H }$ , a capacitor with $C = 0.00125 \mathrm {~F}$ , and a generator producing a voltage of $E ( t ) = 48 \cos ( 10 t )$ If the initial charge is $Q = 0.001 \mathrm { C }$ and the initial current is 0, find the charge $Q ( t )$ at time t.

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

Question 18

A spring with a mass of 2 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 m/s.

A)
B)
C)

Question 19

The figure shows a pendulum with length L and the angle $\theta$ from the vertical to the pendulum. It can be shown that $\theta$ , as a function of time, satisfies the nonlinear differential equation $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + \frac { g } { L } \sin \theta = 0$ where $g = 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 }$ $\text { is the acceleration due to gravity. For small values of }$ $\theta$ we can use the linear approximation $\sin \theta = \theta$ $\text { and then the differential equation becomes linear. Find the equation }$ $\text { of motion of a pendulum with length } 1 \mathrm {~m} \text { if } \theta \text { is initially } 0.2 \mathrm { rad } \text { and the initial }$ $\text { angular velocity is }$ $\frac { d \theta } { d t } = 1 \mathrm { rad } / \mathrm { s }$

A) $$\theta ( t ) = 0.2 \cos ( \sqrt { 9.8 } t ) + \frac { 1 } { \sqrt { 9.8 } } \sin ( \sqrt { 9.8 } t )$$
B) $\theta ( t ) = 0.2 \cos ( \sqrt { 9.8 } t ) + 2 \sin ( \sqrt { 9.8 } t )$
C) $\theta ( t ) = 2 \cos ( 9.8 t ) + \frac { 1 } { 9.8 } \sin ( 9.8 t )$
D) $\theta ( t ) = \frac { 1 } { 9.8 } \cos ( \sqrt { 9.8 } t ) + 0.2 \sin ( \sqrt { 9.8 } t )$
E) $\theta ( t ) = 0.2 \sin ( \sqrt { 9.8 } t ) + \frac { 1 } { \sqrt { 9.8 } } \cos ( \sqrt { 9.8 } t )$

Question 20

Use power series to solve the differential equation. $y ^ { \prime \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { \prime } ( 0 ) = 0$

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 21

Solve the differential equation using the method of variation of parameters.

Question 22

Solve the differential equation using the method of undetermined coefficients.

Question 23

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

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

Question 24

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

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

Question 25

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }$

A) $y ( x ) = c _ { 1 } + \left[ c _ { 2 } + \ln ( \sin x ) \right] \sin x$
B) $y ( x ) = \left( c _ { 1 } + x \right) \sin x + c _ { 2 } + \ln ( \cos x )$
C) $y ( x ) = \frac { \pi } { 2 } \sin x + \left[ \frac { \pi } { 2 } + \ln ( \cos x ) \right] \cos x$
D) $y ( x ) = \left[ \frac { \pi } { 4 } + \ln ( \cos x ) \right] \cos x$
E) $y ( x ) = \left( c _ { 1 } + x \right) \sin x + \left[ c _ { 2 } + \ln ( \cos x ) \right] \cos x$

Question 26

Solve the differential equation.

Question 27

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

A) $\begin{array} { l } y _ { p 1 } ( x ) = A \\y _ { p 2 } ( x ) = B x e ^ { 4 x }\end{array}$
B) $\begin{array} { l } y _ { p 1 } ( x ) = A x \\y _ { p 2 } ( x ) = x ( A x + B ) e ^ { - 4 x }\end{array}$
C) $\begin{array} { l } y _ { 71 } ( x ) = A \\y _ { p 2 } ( x ) = x ( A + B ) e ^ { - 4 x }\end{array}$
D) $\begin{array} { l } y _ { p 1 } ( x ) = A x \\y _ { p 2 } ( x ) = x ^ { 2 } ( A x + B ) e ^ { - 4 x }\end{array}$
E) $\begin{array} { l } y _ { p 1 } ( x ) = A \\y _ { p 2 } ( x ) = x ( A + B x ) e ^ { - 4 x }\end{array}$

Question 28

Solve the initial-value problem using the method of undetermined coefficients.

Question 29

Solve the differential equation using the method of variation of parameters.

Question 30

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

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

Question 31

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

A)
B)
C)

Question 32

Solve the differential equation using the method of variation of parameters.

Question 33

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

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

Question 34

Solve the differential equation.

Question 35

Solve the initial-value problem using the method of undetermined coefficients.

Question 36

Solve the differential equation using the method of variation of parameters.

Question 37

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

A) $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 2 } + C \right) e ^ { - 9 x }$
B) $y _ { y } ( x ) = \left( A x ^ { 3 } + B x ^ { 2 } + C x + D \right) e ^ { 9 x }$
C) $y _ { y } ( x ) = \left( A x ^ { 4 } + B \right) e ^ { 9 x }$
D) $y _ { y } ( x ) = A x ^ { 9 }$
E) $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 3 } + C x ^ { 2 } + D x + E \right) e ^ { 9 x }$

Question 38

Solve the differential equation using the method of undetermined coefficients.

Question 39

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - y ^ { \prime \prime } = e ^ { 2 x }$

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

Question 40

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

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

Question 41

Solve the differential equation.

Question 42

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

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

Question 43

Solve the boundary-value problem, if possible.

Question 44

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

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

Question 45

Solve the differential equation. $4 y ^ { \prime \prime } + y = 0$

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

Question 46

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 5 y ^ { \prime } - 36 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

A) $y ( x ) = \left( e ^ { 8 } - e ^ { - 18 } \right) ^ { - 1 } \left( e ^ { 4 x } - e ^ { - 9 x } \right)$
B) $y ( x ) = \left( e ^ { 4 } - e ^ { - 18 } \right) \left( e ^ { 4 x } - e ^ { - 9 x } \right)$
C) $y ( x ) = \left( e ^ { 9 } - e ^ { - 18 } \right) ^ { - 1 } \left( e ^ { x } - e ^ { - 9 x } \right)$
D) $y ( x ) = \left( e ^ { 4 } - e ^ { - 18 } \right) ^ { - 1 } \left( e ^ { 4 x } - e ^ { - 9 x } \right)$
E) No solution

Question 47

Solve the initial-value problem.

Question 48

Solve the differential equation.

Question 49

Solve the differential equation.

Question 50

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 3 y = 0$

A) $y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$
B) $y ( t ) = e ^ { \frac { \sqrt { 2 } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$
C) $y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$
D) $y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 11 t } { 2 } } \right) \right]$
E) $y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right)$

Question 51

Solve the differential equation. $y ^ { \prime \prime } + 5 y ^ { \prime } - 6 y = 0$

A) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$
B) $y = C _ { 1 } e ^ { - 6 x } + C _ { 2 } e ^ { - 6 x }$
C) $y = \frac { C _ { 1 } e ^ { x } } { 3 } + \frac { C _ { 2 } e ^ { - 6 x } } { 2 }$
D) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 6 x }$
E) None of these

Question 52

Solve the initial-value problem. $y ^ { \prime \prime } - 2 y ^ { \prime } - 24 y = 0 , y ( 1 ) = 4 , y ^ { \prime } ( 1 ) = 5$ .

A) $y ( x ) = \frac { 1 } { 10 } e ^ { 4 ( x - 1 ) } + \frac { 1 } { 23 } e ^ { - 6 ( x - 1 ) }$
B) $y ( x ) = \frac { 1 } { 10 } e ^ { 4 x } - \frac { 1 } { 23 } e ^ { - 6 x }$
C) $y ( x ) = e ^ { 6 x } - e ^ { - 4 x }$
D) $y ( x ) = \frac { 21 } { 10 } e ^ { 6 ( x - 1 ) } + \frac { 19 } { 10 } e ^ { - 4 ( x - 1 ) }$
E) $y ( x ) = \frac { 1 } { 10 } e ^ { x } - \frac { 1 } { 23 } e ^ { - x }$

Question 53

Solve the initial-value problem.

Question 54

Find f by solving the initial value problem. $f ^ { \prime \prime } ( x ) = 8 x ^ { 2 } + 6 x + 4$ ; $f ( - 1 ) = - 7$ , $f ^ { \prime } ( - 1 ) = - 3$

A) $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 10
B) $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 8$
C) $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 16$
D) $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 5

Question 55

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 16 y ^ { \prime } + 64 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 4$

A) $y ( x ) = 4 x e ^ { ( - 8 x + 8 ) }$
B) $y ( x ) = 4 e ^ { ( - 8 x - 8 ) }$
C) $y ( x ) = 4 x e ^ { ( - 8 x - 8 ) }$
D) $y ( x ) = 8 x e ^ { ( - 4 x + 4 ) }$
E) No solution

Question 56

Solve the initial-value problem. $y ^ { \prime \prime } + 8 y ^ { \prime } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 7$

A) $y = e ^ { 5 x } \left( \cos ( 5 x ) + \frac { 11 } { 5 } \sin ( 5 x ) \right)$
B) $y = e ^ { - 5 x } \left( \cos ( 5 x ) + \frac { 2 } { 5 } \sin ( 5 x ) \right)$
C) $y = e ^ { 5 x } \left( \cos ( 4 x ) + \frac { 11 } { 5 } \sin ( 4 x ) \right)$
D) $y = e ^ { - 4 x } \left( \cos ( 5 x ) + \frac { 11 } { 5 } \sin ( 5 x ) \right)$
E) $y = e ^ { 4 x } \left( \cos ( 5 x ) - \frac { 2 } { 5 } \sin ( 5 x ) \right)$

Question 57

Solve the initial-value problem. $$y ^ { \prime \prime } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { \prime } \left( \frac { \pi } { 9 } \right) = 8$$

A) $y ( x ) = - x \sin ( 9 x )$
B) $y ( x ) = \frac { 8 } { 9 } x \sin ( 9 x )$
C) $y ( x ) = - \frac { 8 } { 9 } x \sin ( 9 x )$
D) $y ( x ) = - \frac { 8 } { 9 } \sin ( 9 x )$
E) $y ( x ) = \frac { 8 } { 9 } \sin ( 9 x )$

Question 58

Solve the initial value problem.

Question 59

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

A) $y = \frac { 1 } { 4 } e ^ { 2 x } + \frac { 7 } { 4 } e ^ { - 2 x }$
B) $y = e ^ { x } + \frac { 1 } { 4 } e ^ { - 3 x }$
C) $y = e ^ { 2 x } + \frac { 1 } { 4 } e ^ { - 2 x }$
D) $y = \frac { 7 } { 4 } e ^ { x } + \frac { 1 } { 4 } e ^ { - 3 x }$
E) None of these

Question 60

Solve the differential equation.

Question 61

Solve the boundary-value problem, if possible.

Question 62

Solve the initial-value problem.

Question 63

Solve the differential equation.