Question 1

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

Accepted Answer

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

Question 2

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } - 4 y ^ { t } = \sin 12 x$$

Accepted Answer

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

Question 3

Solve the differential equation using the method of variation of parameters.
$$y ^ { tt } - 5 y ^ {t } + 4 y = 4 \sin x$$

Accepted Answer

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

Question 4

$$\text { Solve the differential equation using the method of variation of parameters. }$$ $$y ^ {tt } + 10 y ^ { t} + 25 y = \frac { e ^ { - 5 x } } { x ^ { 3 } }$$

Accepted Answer

The answer of \[\text { Solve the differential equation using...

Question 5

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

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 6

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the positior 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

A) 
  
B) 
  
C)

Question 7

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 8

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 9

Solve the differential equation.
$$9 y ^ {tt } = 2 y ^ { t }$$

Accepted Answer

The answer of Solve the differential equation.
\[9 y ^ {tt...

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } x \sin ( 2 x ) \right)$ 
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 12

Solve the differential equation using the method of variation of parameters.
$$y ^ { tt } - 3 y ^ {t} = e ^ { 9 x }$$

Accepted Answer

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

Question 13

Solve the differential equation.
$$9 y ^ { tt } = 2 y ^ { t}$$

Accepted Answer

The answer of Solve the differential equation.
\[9 y ^ {...

Question 14

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

Accepted Answer

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

Question 15

Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } - 4 y ^ { t } = \sin 12 x$$

Accepted Answer

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

Question 16

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

The answer of The solution of the initial-value problem \(x...

Question 17

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

Accepted Answer

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

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

Accepted Answer

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

Question 19

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 20

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

Accepted Answer

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

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } = \sin 12 x$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 5 y ^ {t } + 4 y = 4 \sin x$

$\text { Solve the differential equation using the method of variation of parameters. }$ $y ^ {tt } + 10 y ^ { t} + 25 y = \frac { e ^ { - 5 x } } { x ^ { 3 } }$

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

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the positior 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 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 using the method of undetermined coefficients. $y ^ { tt } - 5 y ^ { t } = \sin 15 x$

Solve the differential equation. $9 y ^ {tt } = 2 y ^ { t }$

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

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

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 3 y ^ {t} = e ^ { 9 x }$

Solve the differential equation. $9 y ^ { tt } = 2 y ^ { t}$

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } = \sin 12 x$

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

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

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

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Exam 17: Second-Order Differential Equations

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 Select the correct answer.

Solve the differential equation using the method of undetermined coefficients. ytt−4yt=sin⁡12xy ^ { tt } - 4 y ^ { t } = \sin 12 xytt−4yt=sin12x

Solve the differential equation using the method of variation of parameters. ytt−5yt+4y=4sin⁡xy ^ { tt } - 5 y ^ {t } + 4 y = 4 \sin xytt−5yt+4y=4sinx

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

A spring with a mass of 2 kg2 \mathrm {~kg}2 kg has damping constant 8 and spring constant 80 . Graph the positior 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 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 using the method of undetermined coefficients. ytt−5yt=sin⁡15xy ^ { tt } - 5 y ^ { t } = \sin 15 xytt−5yt=sin15x

Solve the differential equation. 9ytt=2yt9 y ^ {tt } = 2 y ^ { t }9ytt=2yt

Solve the boundary-value problem, if possible. y′′+10y′+25y=0,y(0)=0,y(1)=9y ^ { \prime \prime } + 10 y ^ { \prime } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 9y′′+10y′+25y=0,y(0)=0,y(1)=9

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

Solve the differential equation using the method of variation of parameters. ytt−3yt=e9xy ^ { tt } - 3 y ^ {t} = e ^ { 9 x }ytt−3yt=e9x

Solve the differential equation. 9ytt=2yt9 y ^ { tt } = 2 y ^ { t}9ytt=2yt

Solve the differential equation using the method of variation of parameters. ytt−4yt+3y=2sin⁡xy ^ { tt } - 4 y ^ { t } + 3 y = 2 \sin xytt−4yt+3y=2sinx

Solve the differential equation using the method of undetermined coefficients. ytt−4yt=sin⁡12xy ^ { tt } - 4 y ^ { t } = \sin 12 xytt−4yt=sin12x

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

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

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

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

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Techniques of Integration

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Vector Functions

Partial Derivatives

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Vector Calculus

Filters

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } = \sin 12 x$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 5 y ^ {t } + 4 y = 4 \sin x$

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

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the positior 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 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 using the method of undetermined coefficients. $y ^ { tt } - 5 y ^ { t } = \sin 15 x$

Solve the differential equation. $9 y ^ {tt } = 2 y ^ { t }$

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

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

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 3 y ^ {t} = e ^ { 9 x }$

Solve the differential equation. $9 y ^ { tt } = 2 y ^ { t}$

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } = \sin 12 x$

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

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

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