Question 1

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

Accepted Answer

A)  $y _ { p } = x ^ { 2 } e ^ { 4 x }$ 
B)  $y _ { p } = x e ^ { 4 x } / 7$ 
C)  $y _ { p } = x e ^ { 4 x } / 5$ 
D)  $y _ { y } = e ^ { 4 x }$ 
E)  $y _ { p } = e ^ { 4 x } / 9$ 
A)  $y _ { p } = x ^ { 2 } e ^ { 4 x }$ 
B)  $y _ { p } = x e ^ { 4 x } / 7$ 
C)  $y _ { p } = x e ^ { 4 x } / 5$ 
D)  $y _ { y } = e ^ { 4 x }$ 
E)  $y _ { p } = e ^ { 4 x } / 9$

Question 2

A particular solution of the differential equation $y ^ { \prime \prime } - 2 y ^ { \prime } + y = \sin x$ 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)  $$y _ { p } = - \cos x / 2$$ 
A)  $y _ { p } = \cos x$ 
B)  $y _ { p} = \sin x$ 
C)  $y _ { p } = \sin x / 2$ 
D)  $y _ { p } = \cos x / 2$ 
E)  $$y _ { p } = - \cos x / 2$$

Question 3

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 4 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { 2 x }$ 
B)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = x e ^ { 2 x }$ 
C)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { - 2 x }$ 
D)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = x e ^ { - 2 x }$ 
E)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = x e ^ { 2 x }$ 
A)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { 2 x }$ 
B)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = x e ^ { 2 x }$ 
C)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { - 2 x }$ 
D)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = x e ^ { - 2 x }$ 
E)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = x e ^ { 2 x }$

Question 4

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

Accepted Answer

A)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t - 16$ 
B)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } + 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$ 
C)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { t } + c _ { 2 } t e ^ { t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { t } + c _ { 2 } t e ^ { t } - 5 t + 16$ 
D)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$ 
E)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$ 
A)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t - 16$ 
B)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } + 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$ 
C)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { t } + c _ { 2 } t e ^ { t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { t } + c _ { 2 } t e ^ { t } - 5 t + 16$ 
D)  $x = \left( c _ { 1 } - c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$ 
E)  $x = \left( c _ { 1 } + c _ { 2 } / 5 \right) e ^ { - t } + c _ { 2 } t e ^ { - t } - 4 t + 69 / 5 , y = c _ { 1 } e ^ { - t } + c _ { 2 } t e ^ { - t } - 5 t + 16$

Question 5

The solution of the system of differential equations $\begin{array} { l } 
\frac { d x } { d t } = x + 2 y \
\frac { d y } { d t } = 4 x + 3 y
\end{array}$ is

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } - 6 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { x }$ 
B)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = x e ^ { - x }$ 
C)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { - x }$ 
D)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = x e ^ { - x }$ 
E)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { x }$ 
A)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { x }$ 
B)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = x e ^ { - x }$ 
C)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { - x }$ 
D)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = x e ^ { - x }$ 
E)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { x }$

Question 8

The solution of the system of differential equations $\begin{array} { l } 
\frac { d x } { d t } = 3 x - y \
\frac { d y } { d t } = 9 x - 3 y
\end{array}$ is

Accepted Answer

A)  $x = \left( c _ { 1 } + 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
B)  $x = \left( 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
C)  $x = \left( c _ { 1 } + 3 c _ { 1 } t \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
D)  $x = \left( c _ { 1 } + 3 c _ { 1 } t + 3 c _ { 2 } \right) , y = c _ { 1 } t + c _ { 2 }$ 
E)  $x = \left( c _ { 1 } - 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
A)  $x = \left( c _ { 1 } + 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
B)  $x = \left( 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
C)  $x = \left( c _ { 1 } + 3 c _ { 1 } t \right) / 9 , y = c _ { 1 } t + c _ { 2 }$ 
D)  $x = \left( c _ { 1 } + 3 c _ { 1 } t + 3 c _ { 2 } \right) , y = c _ { 1 } t + c _ { 2 }$ 
E)  $x = \left( c _ { 1 } - 3 c _ { 1 } t + 3 c _ { 2 } \right) / 9 , y = c _ { 1 } t + c _ { 2 }$

Question 9

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 9 y = \sin ( 3 x )$ is

Accepted Answer

A)  $y _ { p } = A \cos ( 3 x )$ 
B)  $y _ { p } = A x \cos ( 3 x ) + B x \sin ( 3 x )$ 
C)  $y _ { p } = A \sin ( 3 x )$ 
D)  $y _ { p } = A x \cos ( 3 x ) + B \sin ( 3 x )$ 
E)  $y _ { p } = A \cos ( 3 x ) + B \sin ( 3 x )$ 
A)  $y _ { p } = A \cos ( 3 x )$ 
B)  $y _ { p } = A x \cos ( 3 x ) + B x \sin ( 3 x )$ 
C)  $y _ { p } = A \sin ( 3 x )$ 
D)  $y _ { p } = A x \cos ( 3 x ) + B \sin ( 3 x )$ 
E)  $y _ { p } = A \cos ( 3 x ) + B \sin ( 3 x )$

Question 10

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 6 y ^ { \prime } + 13 y = e ^ { - 3 x } \cos ( 2 x )$ is

Accepted Answer

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

Question 11

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 5 y = e ^ { - 2 x } \cos x$ is

Accepted Answer

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

Question 12

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 13

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 4 y = 8 x + 2$ is

Accepted Answer

A)  $y _ { p } = 2 x + 1$ 
B)  $y _ { p } = 8 x + 2$ 
C)  $y _ { p } = 2 x - 1$ 
D)  $y _ { p } = x ^ { 2 } + 3 x$ 
E)  $y _ { p } = 2 x - 3$ 
A)  $y _ { p } = 2 x + 1$ 
B)  $y _ { p } = 8 x + 2$ 
C)  $y _ { p } = 2 x - 1$ 
D)  $y _ { p } = x ^ { 2 } + 3 x$ 
E)  $y _ { p } = 2 x - 3$

Question 14

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

Accepted Answer

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

Question 15

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { x } , y _ { 2 } = e ^ { 5 x }$ 
B)  $y _ { 1 } = e ^ { - x } , y _ { 2 } = e ^ { - 5 x }$ 
C)  $y _ { 1 } = e ^ { 2 x } \cos x , y _ { 2 } = e ^ { 2 x } \sin x$ 
D)  $y _ { 1 } = e ^ { x } \cos ( 2 x ) , y _ { 2 } = e ^ { x } \sin ( 2 x )$ 
E)  $y _ { 1 } = e ^ { - x } \cos ( 2 x ) , y _ { 2 } = e ^ { - 2 x } \sin ( 2 x )$ 
A)  $y _ { 1 } = e ^ { x } , y _ { 2 } = e ^ { 5 x }$ 
B)  $y _ { 1 } = e ^ { - x } , y _ { 2 } = e ^ { - 5 x }$ 
C)  $y _ { 1 } = e ^ { 2 x } \cos x , y _ { 2 } = e ^ { 2 x } \sin x$ 
D)  $y _ { 1 } = e ^ { x } \cos ( 2 x ) , y _ { 2 } = e ^ { x } \sin ( 2 x )$ 
E)  $y _ { 1 } = e ^ { - x } \cos ( 2 x ) , y _ { 2 } = e ^ { - 2 x } \sin ( 2 x )$

Question 16

Which of the following sets of functions are linearly independent on $( 0 , \infty )$ ? Select all that apply.

Accepted Answer

A)  $\left\{ 1 , \sin ^ { 2 } x , \cos ^ { 2 } x \right\}$ 
B)  $\{ 1 , x + 3,2 x \}$ 
C)  $\left\{ \sqrt { x } , x , x ^ { 2 } \right\}$ 
D)  $\left\{ 1 , \tan ^ { 2 } x , \sec ^ { 2 } x \right\}$ 
E)  $\{ 1 / x , x , \ln x \}$ 
A)  $\left\{ 1 , \sin ^ { 2 } x , \cos ^ { 2 } x \right\}$ 
B)  $\{ 1 , x + 3,2 x \}$ 
C)  $\left\{ \sqrt { x } , x , x ^ { 2 } \right\}$ 
D)  $\left\{ 1 , \tan ^ { 2 } x , \sec ^ { 2 } x \right\}$ 
E)  $\{ 1 / x , x , \ln x \}$

Question 17

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 3 x y ^ { \prime } + 4 y = 0$ is

Accepted Answer

A)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$ 
B)  $y = c _ { 1 } x + c _ { 2 } x ^ { 3 }$ 
C)  $y = c _ { 1 } x ^ { ( 3 + \sqrt { 7 } ) / 2 } + c _ { 2 } x ^ { ( 3 - \sqrt { 7 } ) / 2 }$ 
D)  $y = c _ { 1 } x ^ { 2 } + c _ { 2 } x ^ { 2 } \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x \ln x$ 
A)  $y = c _ { 1 } x + c _ { 2 } x ^ { 2 }$ 
B)  $y = c _ { 1 } x + c _ { 2 } x ^ { 3 }$ 
C)  $y = c _ { 1 } x ^ { ( 3 + \sqrt { 7 } ) / 2 } + c _ { 2 } x ^ { ( 3 - \sqrt { 7 } ) / 2 }$ 
D)  $y = c _ { 1 } x ^ { 2 } + c _ { 2 } x ^ { 2 } \ln x$ 
E)  $y = c _ { 1 } x + c _ { 2 } x \ln x$

Question 18

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

Accepted Answer

A)  $y = \cos x$ 
B)  $y = \sin x$ 
C)  $y = e ^ { x }$ 
D)  $y = e ^ { - x }$ 
E)  $y = x \cos x$ 
A)  $y = \cos x$ 
B)  $y = \sin x$ 
C)  $y = e ^ { x }$ 
D)  $y = e ^ { - x }$ 
E)  $y = x \cos x$

Question 19

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ are

Accepted Answer

A)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { 3 x }$ 
B)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = e ^ { - 3 x }$ 
C)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { - x }$ 
D)  $$y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { x }$$ 
E)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { - x }$ 
A)  $y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { 3 x }$ 
B)  $y _ { 1 } = e ^ { - 2 x } , y _ { 2 } = e ^ { - 3 x }$ 
C)  $y _ { 1 } = e ^ { 6 x } , y _ { 2 } = e ^ { - x }$ 
D)  $$y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { x }$$ 
E)  $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { - x }$

Question 20

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 5 y = 0$ is

Accepted Answer

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

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

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 4 y = 0$ are

The solution of the system of differential equations =-6x+5y+t =-5x+4y+1 is

The solution of the system of differential equations =x+2y =4x+3y is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } - 6 y = 0$ are

The solution of the system of differential equations =3x-y =9x-3y is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 9 y = \sin ( 3 x )$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 6 y ^ { \prime } + 13 y = e ^ { - 3 x } \cos ( 2 x )$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 5 y = e ^ { - 2 x } \cos x$ is

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

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 4 y = 8 x + 2$ is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 0$ are

Which of the following sets of functions are linearly independent on $( 0 , \infty )$ ? Select all that apply.

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 3 x y ^ { \prime } + 4 y = 0$ is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ are

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 5 y = 0$ 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′′−3y′−4y=e4xy ^ { \prime \prime } - 3 y ^ { \prime } - 4 y = e ^ { 4 x }y′′−3y′−4y=e4x is

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

Two linearly independent solutions of the differential equation y′′−4y′+4y=0y ^ { \prime \prime } - 4 y ^ { \prime } + 4 y = 0y′′−4y′+4y=0 are

The solution of the system of differential equations =-6x+5y+t =-5x+4y+1 is

The solution of the system of differential equations =x+2y =4x+3y is

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

Two linearly independent solutions of the differential equation y′′−5y′−6y=0y ^ { \prime \prime } - 5 y ^ { \prime } - 6 y = 0y′′−5y′−6y=0 are

The solution of the system of differential equations =3x-y =9x-3y is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y′′+9y=sin⁡(3x)y ^ { \prime \prime } + 9 y = \sin ( 3 x )y′′+9y=sin(3x) is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y′′+6y′+13y=e−3xcos⁡(2x)y ^ { \prime \prime } + 6 y ^ { \prime } + 13 y = e ^ { - 3 x } \cos ( 2 x )y′′+6y′+13y=e−3xcos(2x) is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y′′+4y′+5y=e−2xcos⁡xy ^ { \prime \prime } + 4 y ^ { \prime } + 5 y = e ^ { - 2 x } \cos xy′′+4y′+5y=e−2xcosx is

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

A particular solution of the differential equation y′′+3y′+4y=8x+2y ^ { \prime \prime } + 3 y ^ { \prime } + 4 y = 8 x + 2y′′+3y′+4y=8x+2 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

Two linearly independent solutions of the differential equation y′′−4y′+5y=0y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 0y′′−4y′+5y=0 are

Which of the following sets of functions are linearly independent on (0,∞)( 0 , \infty )(0,∞) ? Select all that apply.

The solution of the differential equation x2y′′−3xy′+4y=0x ^ { 2 } y ^ { \prime \prime } - 3 x y ^ { \prime } + 4 y = 0x2y′′−3xy′+4y=0 is

One solution of the differential equation y′′+y′=0y ^ { \prime \prime } + y ^ { \prime } = 0y′′+y′=0 is y=cos⁡xy = \cos xy=cosx . A second linearly independent solution is

Two linearly independent solutions of the differential equation y′′−5y′+6y=0y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0y′′−5y′+6y=0 are

The solution of the differential equation x2y′′−5xy′+5y=0x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 5 y = 0x2y′′−5xy′+5y=0 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 $y ^ { \prime \prime } - 3 y ^ { \prime } - 4 y = e ^ { 4 x }$ is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 4 y = 0$ are

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } - 6 y = 0$ are

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 9 y = \sin ( 3 x )$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 6 y ^ { \prime } + 13 y = e ^ { - 3 x } \cos ( 2 x )$ is

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation $y ^ { \prime \prime } + 4 y ^ { \prime } + 5 y = e ^ { - 2 x } \cos x$ is

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

A particular solution of the differential equation $y ^ { \prime \prime } + 3 y ^ { \prime } + 4 y = 8 x + 2$ is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 0$ are

Which of the following sets of functions are linearly independent on $( 0 , \infty )$ ? Select all that apply.

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 3 x y ^ { \prime } + 4 y = 0$ is

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

Two linearly independent solutions of the differential equation $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ are

The solution of the differential equation $x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 5 y = 0$ is