Question 1

Solve the differential equation using the method of variation of parameters.
Select the correct answer.
$$y ^ { tt } - y ^ { t } = e ^ { 2 x }$$

Accepted Answer

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

Question 2

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

Accepted Answer

The answer of Graph the particular solution and several other...

Question 3

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

Accepted Answer

A) 
$y ( x ) = c _ { 1 } \cos \left( - \frac { x } { 25 } \right) + c _ { 2 } \sin \left( - \frac { x } { 5 } \right)$ 
B)  $y ( x ) = c _ { 1 } \cos ( - 5 x ) + c _ { 2 } \sin ( - 5 x )$ 
C) $y ( x ) = c _ { 1 } \cos \left( \frac { x } { 5 } \right) + c _ { 2 } \sin \left( \frac { x } { 5 } \right)$ 
D)  $y ( x ) = c _ { 1 } \cos ( 25 x ) - c _ { 2 } \sin ( 25 x )$ 
E)  $y ( x ) = c _ { 1 } \cos ( 5 x ) + c _ { 2 } \sin ( 5 x )$ 
A) 
$y ( x ) = c _ { 1 } \cos \left( - \frac { x } { 25 } \right) + c _ { 2 } \sin \left( - \frac { x } { 5 } \right)$ 
B)  $y ( x ) = c _ { 1 } \cos ( - 5 x ) + c _ { 2 } \sin ( - 5 x )$ 
C) $y ( x ) = c _ { 1 } \cos \left( \frac { x } { 5 } \right) + c _ { 2 } \sin \left( \frac { x } { 5 } \right)$ 
D)  $y ( x ) = c _ { 1 } \cos ( 25 x ) - c _ { 2 } \sin ( 25 x )$ 
E)  $y ( x ) = c _ { 1 } \cos ( 5 x ) + c _ { 2 } \sin ( 5 x )$

Question 4

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

Accepted Answer

A)  $y ( x ) = 8 x e ^ { ( - 5 x + 5 ) }$ 
B)  $y ( x ) = 5 e ^ { ( - 8 x - 8 ) }$ 
C)  $y ( x ) = 5 x e ^ { ( - 8 x + 8 ) }$ 
D)  $y ( x ) = 5 x e ^ { ( - 8 x - 8 ) }$ 
E)  No solution 
A)  $y ( x ) = 8 x e ^ { ( - 5 x + 5 ) }$ 
B)  $y ( x ) = 5 e ^ { ( - 8 x - 8 ) }$ 
C)  $y ( x ) = 5 x e ^ { ( - 8 x + 8 ) }$ 
D)  $y ( x ) = 5 x e ^ { ( - 8 x - 8 ) }$ 
E)  No solution

Question 5

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 14 , and a force of $3.6 \mathrm {~N}$ is required to keep the spring stretched $0.3 \mathrm {~m}$ beyond its natural length. The spring is stretched $1 \mathrm {~m}$ beyond its natural length 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 mass of \(2...

Question 6

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

Accepted Answer

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

Question 7

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

The answer of A spring with a \(3 - \mathrm...

Question 8

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 9

Find $f$ by solving the initial value problem.
$$f ^ {tt } ( x ) = 8 x ^ { 2 } + 12 x + 4 ; \quad f ( - 1 ) = - 4 , \quad f ^ { t } ( - 1 ) = - 6$$

Accepted Answer

The answer of Find $f$ by solving the initial value...

Question 10

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 $F ( t ) = 4 F _ { 0 } \cos ( \omega t )$ 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) 
$$x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 2 F _ { 0 } t } { M \omega } \sin ( \omega t )$$ 
C) 
$$x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$$ 
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) 
$$x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 2 F _ { 0 } t } { M \omega } \sin ( \omega t )$$ 
C) 
$$x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$$ 
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 11

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 12

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 13

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } - 4 y ^ { t } + 5 y = 8 e ^ { - x }$$

Accepted Answer

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

Question 14

Solve the initial-value problem.
$$y ^ { tt } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 9 } \right) = 6$$

Accepted Answer

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

Question 15

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

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 16

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

Accepted Answer

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

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 $t$.

Accepted Answer

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

Question 18

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 19

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

Accepted Answer

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

Question 20

Solve the differential equatic
$$25 y ^ { tt } + y = 0$$

Accepted Answer

The answer of Solve the differential equatic
\[25 y ^ {...

Solve the differential equation using the method of variation of parameters. Select the correct answer. $y ^ { tt } - y ^ { t } = e ^ { 2 x }$

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

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

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

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

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

Find $f$ by solving the initial value problem. $f ^ {tt } ( x ) = 8 x ^ { 2 } + 12 x + 4 ; \quad f ( - 1 ) = - 4 , \quad f ^ { t } ( - 1 ) = - 6$

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } + 5 y = 8 e ^ { - x }$

Solve the initial-value problem. $y ^ { tt } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 9 } \right) = 6$

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

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

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

Solve the differential equatic $25 y ^ { tt } + y = 0$

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Exam 17: Second-Order Differential Equations

Solve the differential equation using the method of variation of parameters. Select the correct answer. ytt−yt=e2xy ^ { tt } - y ^ { t } = e ^ { 2 x }ytt−yt=e2x

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 differential equation. Select the correct answer. 25ytt+y=025 y ^ { tt } + y = 025ytt+y=0

Solve the boundary-value problem, if possible. Select the correct answer. 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

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

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

Find fff by solving the initial value problem. ftt(x)=8x2+12x+4;f(−1)=−4,ft(−1)=−6f ^ {tt } ( x ) = 8 x ^ { 2 } + 12 x + 4 ; \quad f ( - 1 ) = - 4 , \quad f ^ { t } ( - 1 ) = - 6ftt(x)=8x2+12x+4;f(−1)=−4,ft(−1)=−6

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

Solve the differential equation using the method of undetermined coefficients. ytt−4yt+5y=8e−xy ^ { tt } - 4 y ^ { t } + 5 y = 8 e ^ { - x }ytt−4yt+5y=8e−x

Solve the initial-value problem. ytt+81y=0,y(π9)=0,yt(π9)=6y ^ { tt } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 9 } \right) = 6ytt+81y=0,y(9π​)=0,yt(9π​)=6

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

Solve the differential equation. 8ytt+yt=08 y ^ { t t } + y ^ { t } = 08ytt+yt=0

Solve the initial-value problem. ytt−2yt−24y=0,y(1)=4,yt(1)=6y ^ {t t } - 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 equatic 25ytt+y=025 y ^ { tt } + y = 025ytt+y=0