Deck 4: Higher-Order Differential Equations

Question 1

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

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

Answer 1

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

Question 2

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

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) $y _ { p } = e ^ { - x } / 4$

Answer 2

$y _ { p } = x ^ { 2 } e ^ { - x } / 2$

Question 3

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

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) $y _ { 1 } = e ^ { - 6 x } , y _ { 2 } = e ^ { - x }$

Answer 3

$y _ { 1 } = e ^ { 2 x } , y _ { 2 } = e ^ { 3 x }$

Question 4

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

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 5

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

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 6

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

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 7

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

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 8

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

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

Question 9

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

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) $y _ { p } = e ^ { 2 x } / 16$

Question 10

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

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 11

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

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 12

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

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 13

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

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 14

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

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

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

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 16

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

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) $m ^ { 2 } + 4 m + 4 = 6$

Question 17

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

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 18

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

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 19

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

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

Question 20

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

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 21

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) $y _ { p } = A e ^ { - 2 x } \cos x$
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 22

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

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 23

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

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 24

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

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 25

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

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 26

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

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 27

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

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 28

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

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 29

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

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 30

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

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 31

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

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 32

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

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 33

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

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 34

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

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

Question 35

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

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 36

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

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 37

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

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 38

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

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) $y _ { p } = e ^ { - x } / 5$

Question 39

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

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

Question 40

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

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) $x = c _ { 1 } e ^ { 5 t } / 2 - c _ { 2 } e ^ { - t } , y = c _ { 1 } e ^ { 5 t } + c _ { 2 } e ^ { - t }$