Question 1

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

Accepted Answer

A) 
  
B) 
  
C)

Question 2

Solve the differential equation. Select the correct answer.
$$y ^ {tt } - 8 y ^ { t } + 41 y = 0$$

Accepted Answer

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

Question 3

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 ^ { tt...

Question 4

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.
Select the correct answer.

Accepted Answer

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) 
$x ( t ) = \frac { F _ { 0 } t } { 2 M \omega } \left( c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) \right)$ 
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 5

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

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 6

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 14 , and a force of $4.8 \mathrm {~N}$ is required to keep the spring stretched $0.4 \mathrm {~m}$ beyond its natural length. Find the mass that would produce critical damping.

Accepted Answer

The answer of A spring with a mass of \(2...

Question 7

Use power series to solve the differential equation.
$$( x - 12 ) y ^ { t } + 2 y = 0$$

Accepted Answer

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

Question 8

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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Question 12

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 13

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 14

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 15

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

The answer of A spring with a mass of \(2...

Question 16

Solve the boundary-value problem, if possible.
$$y ^ { tt } + 5 y ^ {t } - 50 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$$

Accepted Answer

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

Question 17

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

Accepted Answer

The answer of A spring has a mass of \(1...

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

A series circuit consists of a resistor $R = 96 \Omega$, an inductor with $L = 8 H$, 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 $t$.

Accepted Answer

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

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

Solve the differential equation. Select the correct answer. $y ^ {tt } - 8 y ^ { t } + 41 y = 0$

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

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

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 14 , and a force of $4.8 \mathrm {~N}$ is required to keep the spring stretched $0.4 \mathrm {~m}$ beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation. $( x - 12 ) y ^ { t } + 2 y = 0$

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

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

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

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

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

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 $2 \mathrm {~m} / \mathrm { s }$ .

Solve the boundary-value problem, if possible. $y ^ { tt } + 5 y ^ {t } - 50 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

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

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

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

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Exam 17: Second-Order Differential Equations

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

Solve the differential equation. Select the correct answer. ytt−8yt+41y=0y ^ {tt } - 8 y ^ { t } + 41 y = 0ytt−8yt+41y=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

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

A spring with a mass of 2 kg2 \mathrm {~kg}2 kg has damping constant 14 , and a force of 4.8 N4.8 \mathrm {~N}4.8 N is required to keep the spring stretched 0.4 m0.4 \mathrm {~m}0.4 m beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation. (x−12)yt+2y=0( x - 12 ) y ^ { t } + 2 y = 0(x−12)yt+2y=0

Use power series to solve the differential equation.. (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 differential equation using the method of variation of parameters. ytt+y=sec⁡x,π4<x<π2y ^ { tt } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }ytt+y=secx,4π​<x<2π​

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

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 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 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

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 .

Solve the boundary-value problem, if possible. ytt+5yt−50y=0,y(0)=0,y(2)=1y ^ { tt } + 5 y ^ {t } - 50 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1ytt+5yt−50y=0,y(0)=0,y(2)=1

A spring has a mass of 1 kg1 \mathrm {~kg}1 kg and its damping constant is c=10c = 10c=10 . The spring starts from its equilibrium position with a velocity of 1 m/s1 \mathrm {~m} / \mathrm { s }1 m/s . Graph the position function for the spring constant k=20k = 20k=20 .

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

Solve the initial-value problem. 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

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

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

Solve the differential equation. Select the correct answer. $y ^ {tt } - 8 y ^ { t } + 41 y = 0$

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

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

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 14 , and a force of $4.8 \mathrm {~N}$ is required to keep the spring stretched $0.4 \mathrm {~m}$ beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation. $( x - 12 ) y ^ { t } + 2 y = 0$

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

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

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

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

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

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 $2 \mathrm {~m} / \mathrm { s }$ .

Solve the boundary-value problem, if possible. $y ^ { tt } + 5 y ^ {t } - 50 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

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

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

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