Question 1

A particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = e ^ { 2 x }$ is

Accepted Answer

A)  $y _ { p } = e ^ { 2 x }$ 
B)  $y _ { p } = x ^ { 2 } e ^ { 2 x } / 2$ 
C)  $y _ { p } = 2 x e ^ { 2 x }$ 
D)  $y _ { p } = e ^ { 2 x } / 4$ 
E)  $y _ { p } = e ^ { 2 x } / 16$ 
A)  $y _ { p } = e ^ { 2 x }$ 
B)  $y _ { p } = x ^ { 2 } e ^ { 2 x } / 2$ 
C)  $y _ { p } = 2 x e ^ { 2 x }$ 
D)  $y _ { p } = e ^ { 2 x } / 4$ 
E)  $y _ { p } = e ^ { 2 x } / 16$ E

Question 2

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = e ^ { x }$ is

Accepted Answer

A)  $y _ { p } = x ^ { 2 } e ^ { x }$ 
B)  $y _ { p } = x e ^ { x } / 5$ 
C)  $y _ { p } = x e ^ { x }$ 
D)  $y _ { p } = e ^ { x }$ 
E)  $y _ { p } = e ^ { - x } / 5$ 
A)  $y _ { p } = x ^ { 2 } e ^ { x }$ 
B)  $y _ { p } = x e ^ { x } / 5$ 
C)  $y _ { p } = x e ^ { x }$ 
D)  $y _ { p } = e ^ { x }$ 
E)  $y _ { p } = e ^ { - x } / 5$ B

Question 3

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = \cos x$ is

Accepted Answer

A)  $y _ { p } = \cos x$ 
B)  $y _ { p } = \sin x$ 
C)  $y _ { p } = \sin x / 2$ 
D)  $y _ {p } = \cos x / 2$ 
E)  $y _ { p } = - \sin x / 2$ 
A)  $y _ { p } = \cos x$ 
B)  $y _ { p } = \sin x$ 
C)  $y _ { p } = \sin x / 2$ 
D)  $y _ {p } = \cos x / 2$ 
E)  $y _ { p } = - \sin x / 2$ E

Question 4

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 6$ is

Accepted Answer

A)  $m ^ { 2 } + 3 m + 1$ 
B)  $m ^ { 2 } + 3 m + 1 = 0$ 
C)  $m ^ { 2 } + 2 m + 1 = 0$ 
D)  $m ^ { 2 } + 3 m + 1 = 6$ 
E)  $m ^ { 2 } + 2 m + 1 = 6$ 
A)  $m ^ { 2 } + 3 m + 1$ 
B)  $m ^ { 2 } + 3 m + 1 = 0$ 
C)  $m ^ { 2 } + 2 m + 1 = 0$ 
D)  $m ^ { 2 } + 3 m + 1 = 6$ 
E)  $m ^ { 2 } + 2 m + 1 = 6$

Question 5

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y = \cos ( 2 x )$ is

Accepted Answer

A)  $y _ { p } = A \cos ( 2 x )$ 
B)  $y _ { p } = A \cos ( 2 x ) + B \sin ( 2 x )$ 
C)  $y _ { p } = A x \cos ( 2 x )$ 
D)  $y _ { p } = A x \cos ( 2 x ) + B \sin ( 2 x )$ 
E)  $y _ { p } = A x \cos ( 2 x ) + B x \sin ( 2 x )$ 
A)  $y _ { p } = A \cos ( 2 x )$ 
B)  $y _ { p } = A \cos ( 2 x ) + B \sin ( 2 x )$ 
C)  $y _ { p } = A x \cos ( 2 x )$ 
D)  $y _ { p } = A x \cos ( 2 x ) + B \sin ( 2 x )$ 
E)  $y _ { p } = A x \cos ( 2 x ) + B x \sin ( 2 x )$

Question 6

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 9 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = x e ^ { 3 x }$ 
B)  $y _ { 1 } = e ^ { - 3 x } , y _ { 2 } = x e ^ { - 3 x }$ 
C)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = x e ^ { - 3 x }$ 
D)  $y _ { 1 } = e ^ { 3 x } \cos ( x ) , y _ { 2 } = x e ^ { 3 x } \sin ( x )$ 
E)  $y _ { 1 } = e ^ { - 3 x } \cos ( 3 x ) , y _ { 2 } = e ^ { - 3 x } \sin ( x )$ 
A)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = x e ^ { 3 x }$ 
B)  $y _ { 1 } = e ^ { - 3 x } , y _ { 2 } = x e ^ { - 3 x }$ 
C)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = x e ^ { - 3 x }$ 
D)  $y _ { 1 } = e ^ { 3 x } \cos ( x ) , y _ { 2 } = x e ^ { 3 x } \sin ( x )$ 
E)  $y _ { 1 } = e ^ { - 3 x } \cos ( 3 x ) , y _ { 2 } = e ^ { - 3 x } \sin ( x )$

Question 7

The solution of the initial value problem $2 y ^ { \prime \prime } = 3 y ^ { 2 } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 1$ is

Accepted Answer

A)  $y = 2 / ( x + 2 ) ^ { 2 }$ 
B)  $y = 4 / ( x + 2 ) ^ { 3 }$ 
C)  $y = 1 / ( x + 1 ) ^ { 3 }$ 
D)  $y = 1 / ( x + 1 ) ^ { 2 }$ 
E)  $y = 4 / ( x + 2 ) ^ { 2 }$ 
A)  $y = 2 / ( x + 2 ) ^ { 2 }$ 
B)  $y = 4 / ( x + 2 ) ^ { 3 }$ 
C)  $y = 1 / ( x + 1 ) ^ { 3 }$ 
D)  $y = 1 / ( x + 1 ) ^ { 2 }$ 
E)  $y = 4 / ( x + 2 ) ^ { 2 }$

Question 8

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 25 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = e ^ { 4 x }$ 
B)  $y _ { 1 } = e ^ { - 3 x } , y _ { 2 } = e ^ { - 4 x }$ 
C)  $y _ { 1 } = e ^ { - 3 x } \cos ( 4 x ) , y _ { 2 } = e ^ { - 3 x } \sin ( 4 x )$ 
D)  $y _ { 1 } = e ^ { 3 x } \cos ( 4 x ) , y _ { 2 } = e ^ { 3 x } \sin ( 4 x )$ 
E)  $y _ { 1 } = e ^ { 4 x } \cos ( 3 x ) , y _ { 2 } = e ^ { 4 x } \sin ( 3 x )$ 
A)  $y _ { 1 } = e ^ { 3 x } , y _ { 2 } = e ^ { 4 x }$ 
B)  $y _ { 1 } = e ^ { - 3 x } , y _ { 2 } = e ^ { - 4 x }$ 
C)  $y _ { 1 } = e ^ { - 3 x } \cos ( 4 x ) , y _ { 2 } = e ^ { - 3 x } \sin ( 4 x )$ 
D)  $y _ { 1 } = e ^ { 3 x } \cos ( 4 x ) , y _ { 2 } = e ^ { 3 x } \sin ( 4 x )$ 
E)  $y _ { 1 } = e ^ { 4 x } \cos ( 3 x ) , y _ { 2 } = e ^ { 4 x } \sin ( 3 x )$

Question 9

A solution of the equation for the differential equation $y ^ { \prime \prime } = 2 x \left( y ^ { \prime } \right) ^ { 2 }$ is

Accepted Answer

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

Question 10

The solution of the system of differential equations $\begin{array} { l } 
\frac { d x } { d t } = - 6 x + 5 y \
\frac { d y } { d t } = - 5 x + 4 y
\end{array}$ is

Accepted Answer

A)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { t } + c _ { 2 } t e ^ { t } , y = c _ { 1 } e ^ { t } + c _ { 2 } t e ^ { t }$ 
B)  $x = \left( c _ { 1 } - c _ { 2 } \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
C)  $x = \left( c _ { 1 } + c _ { 2 } \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
D)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
E)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
A)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { t } + c _ { 2 } t e ^ { t } , y = c _ { 1 } e ^ { t } + c _ { 2 } t e ^ { t }$ 
B)  $x = \left( c _ { 1 } - c _ { 2 } \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
C)  $x = \left( c _ { 1 } + c _ { 2 } \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
D)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$ 
E)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t }$

Question 11

The solution of the differential equation $y ^ { \prime } y ^ { \prime \prime } = 4$ is

Accepted Answer

A)  $y = \left( 8 x + c _ { 1 } \right) ^ { 3 / 2 } / 24 + c _ { 2 }$ 
B)  $y = \left( 4 x + c _ { 1 } \right) ^ { 3 / 2 } / 12 + c _ { 2 }$ 
C)  $y = \left( 4 x + c _ { 1 } \right) ^ { 3 / 2 } / 24 + c _ { 2 }$ 
D)  $y = \left( 8 x + c _ { 1 } \right) ^ { 1 / 2 } / 12 + c _ { 2 }$ 
E)  $y = \left( 8 x + c _ { 1 } \right) ^ { 3 / 2 } / 12 + c _ { 2 }$ 
A)  $y = \left( 8 x + c _ { 1 } \right) ^ { 3 / 2 } / 24 + c _ { 2 }$ 
B)  $y = \left( 4 x + c _ { 1 } \right) ^ { 3 / 2 } / 12 + c _ { 2 }$ 
C)  $y = \left( 4 x + c _ { 1 } \right) ^ { 3 / 2 } / 24 + c _ { 2 }$ 
D)  $y = \left( 8 x + c _ { 1 } \right) ^ { 1 / 2 } / 12 + c _ { 2 }$ 
E)  $y = \left( 8 x + c _ { 1 } \right) ^ { 3 / 2 } / 12 + c _ { 2 }$

Question 12

The solution of the initial value problem $\begin{array} { l } 
\frac { d x } { d t } = 10 x - 5 y \
\frac { d y } { d t } = 8 x - 12 y \
x ( 0 ) = 2 , y ( 0 ) = 1
\end{array}$ is

Accepted Answer

A)  $x = \left( 35 e ^ { 8 t } - e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
B)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
C)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 9 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
D)  $x = \left( 35 e ^ { 8 t } - e ^ { - 10 t } \right) / 9 , y = \left( 7 e ^ { 8 t } - 2 e ^ { - 10 t } \right) / 9$ 
E)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } - 2 e ^ { - 10 t } \right) / 9$ 
A)  $x = \left( 35 e ^ { 8 t } - e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
B)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
C)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 9 , y = \left( 7 e ^ { 8 t } + 2 e ^ { - 10 t } \right) / 9$ 
D)  $x = \left( 35 e ^ { 8 t } - e ^ { - 10 t } \right) / 9 , y = \left( 7 e ^ { 8 t } - 2 e ^ { - 10 t } \right) / 9$ 
E)  $x = \left( 35 e ^ { 8 t } + e ^ { - 10 t } \right) / 18 , y = \left( 7 e ^ { 8 t } - 2 e ^ { - 10 t } \right) / 9$

Question 13

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 2 y = 4 x + 3$ is

Accepted Answer

A)  $y _ { p } = 4 x + 3$ 
B)  $y _ { p } = 2 x + 3 / 2$ 
C)  $y _ { p } = 2 x - 3 / 2$ 
D)  $y _ { p } = 4 x ^ { 2 } + 3 x$ 
E)  $y _ { p } = 2 x - 3$ 
A)  $y _ { p } = 4 x + 3$ 
B)  $y _ { p } = 2 x + 3 / 2$ 
C)  $y _ { p } = 2 x - 3 / 2$ 
D)  $y _ { p } = 4 x ^ { 2 } + 3 x$ 
E)  $y _ { p } = 2 x - 3$

Question 14

The Wronskian of the functions $e ^ { x }$ and $e ^ { 3 x }$ is

Accepted Answer

A)  $2 e ^ { 4 x }$ 
B)  $- 2 e ^ { 4 x }$ 
C)  $2 e ^ { 2 x }$ 
D)  $4 e ^ { 4 x }$ 
E)  $4 e ^ { 2 x }$ 
A)  $2 e ^ { 4 x }$ 
B)  $- 2 e ^ { 4 x }$ 
C)  $2 e ^ { 2 x }$ 
D)  $4 e ^ { 4 x }$ 
E)  $4 e ^ { 2 x }$

Question 15

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + 2 y = 0$ is

Accepted Answer

A)  $y = c _ { 1 } x \cos ( \ln x ) + c _ { 2 } x \sin ( \ln x )$ 
B)  $y = c _ { 1 } x ^ { 1 / 2 } \cos ( \sqrt { 3 } \ln x / 2 ) + c _ { 2 } x ^ { 1 / 2 } \sin ( \sqrt { 3 } \ln x / 2 )$ 
C)  $y = c _ { 1 } x ^ { ( 1 + \sqrt { 3 } ) / 2 } + c _ { 2 } x ^ { ( 1 - \sqrt { 3 } ) / 2 }$ 
D)  $y = c _ { 1 } x + c _ { 2 } x \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$ 
A)  $y = c _ { 1 } x \cos ( \ln x ) + c _ { 2 } x \sin ( \ln x )$ 
B)  $y = c _ { 1 } x ^ { 1 / 2 } \cos ( \sqrt { 3 } \ln x / 2 ) + c _ { 2 } x ^ { 1 / 2 } \sin ( \sqrt { 3 } \ln x / 2 )$ 
C)  $y = c _ { 1 } x ^ { ( 1 + \sqrt { 3 } ) / 2 } + c _ { 2 } x ^ { ( 1 - \sqrt { 3 } ) / 2 }$ 
D)  $y = c _ { 1 } x + c _ { 2 } x \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$

Question 16

One solution of the differential equation $y ^ { \prime \prime } + y ^ { \prime } = 0$ is $y = e ^ { - x }$ . A second linearly independent solution is

Accepted Answer

A)  $y = c$ 
B)  $y = e ^ { x }$ 
C)  $y = x e ^ { x }$ 
D)  $y = x e ^ { - x }$ 
E)  $y = e ^ { - x }$ 
A)  $y = c$ 
B)  $y = e ^ { x }$ 
C)  $y = x e ^ { x }$ 
D)  $y = x e ^ { - x }$ 
E)  $y = e ^ { - x }$

Question 17

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = e ^ { x }$ is

Accepted Answer

A)  $y _ { p } = x ^ { 2 } e ^ { x } / 2$ 
B)  $y _ { p } = x e ^ { x } / 2$ 
C)  $y _ { p } = x e ^ { x }$ 
D)  $y _ { p } = e ^ { x }$ 
E)  $y _ { p } = e ^ { - x } / 4$ 
A)  $y _ { p } = x ^ { 2 } e ^ { x } / 2$ 
B)  $y _ { p } = x e ^ { x } / 2$ 
C)  $y _ { p } = x e ^ { x }$ 
D)  $y _ { p } = e ^ { x }$ 
E)  $y _ { p } = e ^ { - x } / 4$

Question 18

The solution of the system of differential equations $\begin{array} { l } 
\frac { d x } { d t } = - 6 x + 5 y \
\frac { d y } { d t } = - 5 x + 4 y \
x ( 0 ) = 1 / 3 , y ( 0 ) = 0
\end{array}$ is

Accepted Answer

A)  $x = \left( e ^ { t } - 5 t e ^ { t } \right) / 3 , y = - 5 t e ^ { t } / 3$ 
B)  $x = \left( e ^ { - t } - t e ^ { - t } \right) / 3 , y = - t e ^ { - t } / 3$ 
C)  $x = \left( e ^ { - t } + t e ^ { - t } \right) / 3 , y = t e ^ { - t } / 3$ 
D)  $x = \left( e ^ { - t } + 5 t e ^ { - t } \right) / 3 , y = 5 t e ^ { - t } / 3$ 
E)  $x = \left( e ^ { - t } - 5 t e ^ { - t } \right) / 3 , y = - 5 t e ^ { - t } / 3$ 
A)  $x = \left( e ^ { t } - 5 t e ^ { t } \right) / 3 , y = - 5 t e ^ { t } / 3$ 
B)  $x = \left( e ^ { - t } - t e ^ { - t } \right) / 3 , y = - t e ^ { - t } / 3$ 
C)  $x = \left( e ^ { - t } + t e ^ { - t } \right) / 3 , y = t e ^ { - t } / 3$ 
D)  $x = \left( e ^ { - t } + 5 t e ^ { - t } \right) / 3 , y = 5 t e ^ { - t } / 3$ 
E)  $x = \left( e ^ { - t } - 5 t e ^ { - t } \right) / 3 , y = - 5 t e ^ { - t } / 3$

Question 19

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 0$ is

Accepted Answer

A)  $y = c _ { 1 } x ^ { - 1 } + c _ { 2 } x ^ { - 1 } \ln x$ 
B)  $y = c _ { 1 } x ^ { - 1 } + c _ { 2 } x ^ { - 2 }$ 
C)  $y = c _ { 1 } x ^ { ( - 3 + \sqrt { 5 } ) / 2 } + c _ { 2 } x ^ { ( - 3 - \sqrt { 5 } ) / 2 }$ 
D)  $y = c _ { 1 } x + c _ { 2 } x \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$ 
A)  $y = c _ { 1 } x ^ { - 1 } + c _ { 2 } x ^ { - 1 } \ln x$ 
B)  $y = c _ { 1 } x ^ { - 1 } + c _ { 2 } x ^ { - 2 }$ 
C)  $y = c _ { 1 } x ^ { ( - 3 + \sqrt { 5 } ) / 2 } + c _ { 2 } x ^ { ( - 3 - \sqrt { 5 } ) / 2 }$ 
D)  $y = c _ { 1 } x + c _ { 2 } x \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$

Question 20

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 5 y ^ { \prime } + 4 y = 6$ is

Accepted Answer

A)  $m ^ { 2 } + 5 m + 4$ 
B)  $m ^ { 2 } + 4 m + 4 = 0$ 
C)  $m ^ { 2 } + 5 m + 4 = 0$ 
D)  $m ^ { 2 } + 5 m + 4 = 6$ 
E)  $m ^ { 2 } + 4 m + 4 = 6$ 
A)  $m ^ { 2 } + 5 m + 4$ 
B)  $m ^ { 2 } + 4 m + 4 = 0$ 
C)  $m ^ { 2 } + 5 m + 4 = 0$ 
D)  $m ^ { 2 } + 5 m + 4 = 6$ 
E)  $m ^ { 2 } + 4 m + 4 = 6$

A particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = e ^ { 2 x }$ is

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = e ^ { x }$ is

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = \cos x$ is

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 6$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y = \cos ( 2 x )$ is

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 9 y = 0$ are

The solution of the initial value problem $2 y ^ { \prime \prime } = 3 y ^ { 2 } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 1$ is

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 25 y = 0$ are

A solution of the equation for the differential equation $y ^ { \prime \prime } = 2 x \left( y ^ { \prime } \right) ^ { 2 }$ is

The solution of the system of differential equations =-6x+5y =-5x+4y is

The solution of the differential equation $y ^ { \prime } y ^ { \prime \prime } = 4$ is

The solution of the initial value problem =10x-5y =8x-12y x(0)=2,y(0)=1 is

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 2 y = 4 x + 3$ is

The Wronskian of the functions $e ^ { x }$ and $e ^ { 3 x }$ is

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + 2 y = 0$ is

One solution of the differential equation $y ^ { \prime \prime } + y ^ { \prime } = 0$ is $y = e ^ { - x }$ . A second linearly independent solution is

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = e ^ { x }$ is

The solution of the system of differential equations =-6x+5y =-5x+4y x(0)=1/3,y(0)=0 is

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 0$ is

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 5 y ^ { \prime } + 4 y = 6$ is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 4: Higher-Order Differential Equations

A particular solution of the differential equation y′′+4y′+4y=e2xy ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = e ^ { 2 x }y′′+4y′+4y=e2x is

A particular solution of the differential equation y′′+3y′−4y=exy ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = e ^ { x }y′′+3y′−4y=ex is

A particular solution of the differential equation y′′−2y′+y=cos⁡xy ^ { \prime \prime } - 2 y ^ { \prime } + y = \cos xy′′−2y′+y=cosx is

The auxiliary equation for the differential equation x2y′′+3xy′+y=6x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 6x2y′′+3xy′+y=6 is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y′′+4y=cos⁡(2x)y ^ { \prime \prime } + 4 y = \cos ( 2 x )y′′+4y=cos(2x) is

Two linearly independent solutions of the differential equation y′′−6y′+9y=0y ^ { \prime \prime } - 6 y ^ { \prime } + 9 y = 0y′′−6y′+9y=0 are

The solution of the initial value problem 2y′′=3y2,y(0)=1,y′(0)=12 y ^ { \prime \prime } = 3 y ^ { 2 } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 12y′′=3y2,y(0)=1,y′(0)=1 is

Two linearly independent solutions of the differential equation y′′−6y′+25y=0y ^ { \prime \prime } - 6 y ^ { \prime } + 25 y = 0y′′−6y′+25y=0 are

A solution of the equation for the differential equation y′′=2x(y′)2y ^ { \prime \prime } = 2 x \left( y ^ { \prime } \right) ^ { 2 }y′′=2x(y′)2 is

The solution of the system of differential equations =-6x+5y =-5x+4y is

The solution of the differential equation y′y′′=4y ^ { \prime } y ^ { \prime \prime } = 4y′y′′=4 is

The solution of the initial value problem =10x-5y =8x-12y x(0)=2,y(0)=1 is

A particular solution of the differential equation y′′+3y′+2y=4x+3y ^ { \prime \prime } + 3 y ^ { \prime } + 2 y = 4 x + 3y′′+3y′+2y=4x+3 is

The Wronskian of the functions exe ^ { x }ex and e3xe ^ { 3 x }e3x is

The solution of the equation for the differential equation x2y′′−2xy′+2y=0x ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + 2 y = 0x2y′′−2xy′+2y=0 is

One solution of the differential equation y′′+y′=0y ^ { \prime \prime } + y ^ { \prime } = 0y′′+y′=0 is y=e−xy = e ^ { - x }y=e−x . A second linearly independent solution is

A particular solution of the differential equation y′′−2y′+y=exy ^ { \prime \prime } - 2 y ^ { \prime } + y = e ^ { x }y′′−2y′+y=ex is

The solution of the system of differential equations =-6x+5y =-5x+4y x(0)=1/3,y(0)=0 is

The solution of the equation for the differential equation x2y′′+3xy′+y=0x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 0x2y′′+3xy′+y=0 is

The auxiliary equation for the differential equation x2y′′+5y′+4y=6x ^ { 2 } y ^ { \prime \prime } + 5 y ^ { \prime } + 4 y = 6x2y′′+5y′+4y=6 is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

A particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = e ^ { 2 x }$ is

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = e ^ { x }$ is

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = \cos x$ is

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 6$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y = \cos ( 2 x )$ is

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 9 y = 0$ are

The solution of the initial value problem $2 y ^ { \prime \prime } = 3 y ^ { 2 } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 1$ is

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 6 y ^ { \prime } + 25 y = 0$ are

A solution of the equation for the differential equation $y ^ { \prime \prime } = 2 x \left( y ^ { \prime } \right) ^ { 2 }$ is

The solution of the differential equation $y ^ { \prime } y ^ { \prime \prime } = 4$ is

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 2 y = 4 x + 3$ is

The Wronskian of the functions $e ^ { x }$ and $e ^ { 3 x }$ is

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + 2 y = 0$ is

One solution of the differential equation $y ^ { \prime \prime } + y ^ { \prime } = 0$ is $y = e ^ { - x }$ . A second linearly independent solution is

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = e ^ { x }$ is

The solution of the equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + y = 0$ is

The auxiliary equation for the differential equation $x ^ { 2 } y ^ { \prime \prime } + 5 y ^ { \prime } + 4 y = 6$ is