Deck 8: Systems of Linear First-Order Differential Equations

Question 1

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right) X$ is

A) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + c _ { 2 } \left[ \left( \begin{array} { c } 1 \\- 1\end{array} \right) t e ^ { - t } + \left( \begin{array} { c } - 1 / 2 \\0\end{array} \right) e ^ { - t } \right]$
B) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { t } + c _ { 2 } \left[ \left( \begin{array} { c } 1 \\- 1\end{array} \right) t e ^ { t } + \left( \begin{array} { c } - 1 / 2 \\0\end{array} \right) e ^ { t } \right]$
C) $X = c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { t }$
D) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { t }$
E) none of the above

Answer 1

$\mathbf { X } = c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + c _ { 2 } \left[ \left( \begin{array} { c } 1 \\- 1\end{array} \right) t e ^ { - t } + \left( \begin{array} { c } - 1 / 2 \\0\end{array} \right) e ^ { - t } \right]$

Question 2

The Wronskian of the vector functions $\mathbf { X } _ { 1 } = \left( \begin{array} { l } 1 \\2 \\3\end{array} \right) e ^ { t } , \mathbf { X } _ { 2 } = \left( \begin{array} { c } - 1 \\0 \\3\end{array} \right) e ^ { - 2 t }$ and $\mathbf { X } _ { 3 } = \left( \begin{array} { l } 2 \\2 \\1\end{array} \right) e ^ { 4 t }$ is

A) $14 e ^ { 3 t }$
B) $10 e ^ { 3 t }$
C) $- 10 e ^ { 3 t }$
D) $2 e ^ { 3 t }$
E) $- 2 e ^ { 3 t }$

Answer 2

$2 e ^ { 3 t }$

Question 3

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }$ is

A) $X = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { - 3 t }$
B) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { 3 t }$
C) $X = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { - 3 t }$
D) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { 3 t }$
E) none of the above

Answer 3

none of the above

Question 4

The eigenvalues of $A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)$ is

A) $- 2 \pm \sqrt { 2 }$
B) $2 \pm \sqrt { 2 }$
C) $- 2 \pm 2 i$
D) $2 \pm 2 i$
E) $2 \pm 2 \sqrt { 3 }$

Question 5

The characteristic equation of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

A) $\lambda ^ { 2 } - 3 \lambda + 4 = 0$
B) $\lambda ^ { 2 } - 3 \lambda - 4 = 0$
C) $\lambda ^ { 2 } - 3 \lambda = 0$
D) $\lambda ^ { 2 } + 3 \lambda = 0$
E) $\lambda ^ { 2 } + 3 \lambda + 4 = 0$

Question 6

If $X _ { 1 }$ and $X _ { 2 }$ are solutions of the second order system $\mathbf { X } ^ { \prime } = A \mathbf { X }$ and $\mathrm { X } _ { p }$ is a particular solution of $\mathrm { X } ^ { \prime } = A \mathrm { X } + f ( t )$ , then the general solution of $\mathrm { X } ^ { \prime } = A \mathrm { X } + f ( t )$ is

A) $\mathrm { X } _ { 1 } + \mathrm { X } _ { 2 } + \mathrm { X } _ { p }$
B) $\mathbf { X } _ { 1 } + \mathbf { X } _ { 2 } + c _ { 3 } \mathbf { X } _ { p }$
C) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 }$
D) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + c _ { 3 } \mathbf { X } _ { p }$
E) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + \mathbf { X } _ { p }$

Question 7

The particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { c } e ^ { - t } \\0\end{array} \right)$ is

A) $\left( \begin{array} { c } 2 t e ^ { t } / 5 + 3 e ^ { t } \\- 2 t e ^ { t } / 5 + 2 e ^ { t }\end{array} \right)$
B) $\left( \begin{array} { c } 2 t e ^ { - t } / 5 + 3 e ^ { - t } \\- 2 t e ^ { - t } / 5 + 2 e ^ { - t }\end{array} \right)$
C) $\left( \begin{array} { c } 2 t e ^ { - t } / 5 - 3 e ^ { - t } \\- 2 t e ^ { - t } / 5 - 2 e ^ { - t }\end{array} \right)$
D) $\left( \begin{array} { c } 2 t e ^ { t } / 5 - 3 e ^ { t } \\- 2 t e ^ { t } / 5 - 2 e ^ { t }\end{array} \right)$
E) none of the above

Question 8

The particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)$ is

A) $\left( \begin{array} { c } t / 4 + 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
B) $\left( \begin{array} { c } t / 4 - 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
C) $\left( \begin{array} { c } t / 4 + 1 / 8 \\- t / 2 - 7 / 8\end{array} \right)$
D) $\left( \begin{array} { c } t / 4 - 1 / 8 \\- t / 2 - 7 / 8\end{array} \right)$
E) none of the above

Question 9

The characteristic equation of $A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)$ is

A) $- \lambda ^ { 3 } + \lambda ^ { 2 } + 25 \lambda - 25 = 0$
B) $- \lambda ^ { 3 } + \lambda ^ { 2 } - 25 \lambda + 25 = 0$
C) $- \lambda ^ { 3 } - \lambda ^ { 2 } + 25 \lambda - 25 = 0$
D) $- \lambda ^ { 3 } - \lambda ^ { 2 } - 25 \lambda + 25 = 0$
E) none of the above

Question 10

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right) X$ is

A) $X = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { - 3 t }$
B) $X = c _ { 1 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - 3 t }$
C) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { 3 t }$
D) $\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { 3 t }$
E) none of the above

Question 11

The eigenvalues of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

A) $1,1$
B) $- 1 , - 1$
C) $1 \pm \sqrt { 5 }$
D) $1 \pm \sqrt { 2 }$
E) $- 1 \pm \sqrt { 2 }$

Question 12

The Wronskian of the vector functions $X _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }$ and $X _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }$ is

A) $8 e ^ { 4 t }$
B) $2 e ^ { 4 t }$
C) $- 8 e ^ { 4 t }$
D) $- 2 e ^ { 4 t }$
E) $15 e ^ { 4 t }$

Question 13

The general solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)$ is

A) $c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { - 4 t } + \left( \begin{array} { c } t / 4 + 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
B) $c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { - 4 t } + \left( \begin{array} { c } t / 4 + 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$ .
C) $c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { 4 t } + \left( \begin{array} { c } t / 4 - 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
D) $c _ { 1 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { - 4 t } + \left( \begin{array} { c } t / 4 + 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$ ..
E) none of the above

Question 14

The eigenvalues of $A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)$ is

A) $1 , \pm 5 i$
B) $- 1 , \pm 5 i$
C) $1,4 , - 6$
D) $1 , - 4,6$
E) none of the above

Question 15

The eigenvalues of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

A) $\pm 1$
B) $- 2 \pm i$
C) $2 \pm i$
D) $1,3$
E) $- 1 , - 3$

Question 16

The characteristic equation of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

A) $\lambda ^ { 2 } - 1 = 0$
B) $\lambda ^ { 2 } + 4 \lambda + 5 = 0$
C) $\lambda ^ { 2 } - 4 \lambda + 5 = 0$
D) $\lambda ^ { 2 } + 4 \lambda + 3 = 0$
E) $\lambda ^ { 2 } - 4 \lambda + 3 = 0$

Question 17

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)$ is

A) $\lambda ^ { 2 } + 4 \lambda - 2 = 0$
B) $\lambda ^ { 2 } - 4 \lambda - 2 = 0$
C) $\lambda ^ { 2 } - 4 \lambda - 8 = 0$
D) $\lambda ^ { 2 } + 4 \lambda + 8 = 0$
E) $\lambda ^ { 2 } + 4 \lambda + 2 = 0$

Question 18

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

A) $0,3$
B) $0 , - 3$
C) $( 3 \pm \sqrt { 7 } i ) / 2$
D) $( - 3 \pm \sqrt { 7 } i ) / 2$
E) $- 1,4$

Question 19

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

A) $\lambda ^ { 2 } - 2 \lambda - 4 = 0$
B) $\lambda ^ { 2 } - 2 \lambda - 1 = 0$
C) $\lambda ^ { 2 } - 2 \lambda + 1 = 0$
D) $\lambda ^ { 2 } + 2 \lambda - 1 = 0$
E) $\lambda ^ { 2 } + 2 \lambda + 1 = 0$

Question 20

The solution of the previous problem that satisfies the initial condition $X ( 0 ) = \left( \begin{array} { l } 0 \\0\end{array} \right)$ is

A) $\left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { 4 t } / 16 + \left( \begin{array} { c } t / 4 - 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
B) $\left(\begin{array}{c}1 \\-1\end{array}\right) e^{-t} / 16+\left(\begin{array}{l}3 \\2\end{array}\right) e^{4 t}+\left(\begin{array}{c}t / 4+19 / 16 \\-t / 2+7 / 8\end{array}\right)$
C) $\left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - t } + \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { 4 t } / 16 + \left( \begin{array} { c } t / 4 + 19 / 16 \\- t / 2 + 7 / 8\end{array} \right)$
D) $\left(\begin{array}{c}1 \\-1\end{array}\right) e^{-t} / 16+\left(\begin{array}{l}3 \\2\end{array}\right) e^{4 t}+\left(\begin{array}{c}t / 4-19 / 16 \\-t / 2+7 / 8\end{array}\right)$
E) none of the above

Question 21

The characteristic equation of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

A) $\lambda ^ { 2 } + 3 \lambda - 4 = 0$
B) $\lambda ^ { 2 } + 3 \lambda + 4 = 0$
C) $\lambda ^ { 2 } - 3 \lambda = 0$
D) $\lambda ^ { 2 } + 3 \lambda = 0$
E) $\lambda ^ { 2 } - 3 \lambda - 4 = 0$

Question 22

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right) \mathbf { X }$ is

A) $\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { - 2 t }$
B) $\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 3 \\2\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { 2 t }$
C) $\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 2 \\3\end{array} \right) e ^ { - t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { - 2 t }$
D) $\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 2 \\3\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { 2 t }$
E) none of the above

Question 23

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }$ is

A) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) + c _ { 2 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { - 3 t }$
B) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) + c _ { 2 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { 3 t }$
C) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { - 3 t }$
D) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { 3 t }$
E) none of the above

Question 24

A particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { c } 1 \\e ^ { t }\end{array} \right)$ is

A) $\left( \begin{array} { c } e ^ { t } / 2 \\- e ^ { t } / 2 + 1\end{array} \right)$
B) $\left( \begin{array} { c } - e ^ { t } / 2 \\e ^ { t } / 2 - 1\end{array} \right)$
C) $\left( \begin{array} { c } e ^ { t } / 2 \\- e ^ { t } / 2 - 1\end{array} \right)$
D) $\left( \begin{array} { c } e ^ { t } / 2 \\e ^ { t } / 2 + 1\end{array} \right)$
E) $\left( \begin{array} { c } e ^ { t } / 2 \\e ^ { t } / 2 - 1\end{array} \right)$

Question 25

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right) \mathbf { X }$ is

A) $\mathbf { X } = c _ { 1 } e ^ { 2 t } \left[ \left( \begin{array} { l } 0 \\1\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { l } 0 \\1\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } e ^ { 2 t } \left[ \left( \begin{array} { l } 1 \\0\end{array} \right) \cos ( 2 t ) + \left( \begin{array} { l } 0 \\1\end{array} \right) \sin ( 2 t ) \right]$
B) $\mathbf { X } = c _ { 1 } e ^ { - 2 t } \left[ \left( \begin{array} { l } 0 \\1\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { l } 1 \\0\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } e ^ { - 2 t } \left[ \left( \begin{array} { l } 1 \\0\end{array} \right) \cos ( 2 t ) + \left( \begin{array} { l } 0 \\1\end{array} \right) \sin ( 2 t ) \right]$
C) $\mathrm { X } = c _ { 1 } e ^ { 2 t } \left[ \left( \begin{array} { l } 0 \\1\end{array} \right) \cos t + \left( \begin{array} { l } 1 \\0\end{array} \right) \sin t \right] + c _ { 2 } e ^ { 2 t } \left[ \left( \begin{array} { l } 1 \\0\end{array} \right) \cos t + \left( \begin{array} { l } 0 \\1\end{array} \right) \sin t \right]$
D) $\mathrm { X } = c _ { 1 } e ^ { - 2 t } \left[ \left( \begin{array} { l } 0 \\1\end{array} \right) \cos t - \left( \begin{array} { l } 1 \\0\end{array} \right) \sin t \right] + c _ { 2 } e ^ { - 2 t } \left[ \left( \begin{array} { l } 1 \\0\end{array} \right) \cos t + \left( \begin{array} { l } 0 \\1\end{array} \right) \sin t \right]$
E) none of the above

Question 26

Let A be the matrix of the previous two problems. The solution of $\mathbf { X } ^ { \prime } = A \mathbf { X }$ is

A) $c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$
B) $c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { - 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { - 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { - 2 t } + \left( \begin{array} { c } 0 \\- 1 \\0\end{array} \right) e ^ { - 2 t } \right]$
C) $c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { - 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { - 2 t } + \left( \begin{array} { c } 0 \\- 1 \\0\end{array} \right) e ^ { - 2 t } \right]$
D) $c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { - 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { c } 0 \\- 1 \\0\end{array} \right) e ^ { 2 t } \right]$
E) $c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { c } 0 \\- 1 \\0\end{array} \right) e ^ { 2 t } \right]$

Question 27

The characteristic equation for the matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)$ is

A) $- \lambda ^ { 3 } - 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$
B) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 14 \lambda + 6 = 0$
C) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 14 \lambda + 6 = 0$
D) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 16 \lambda + 12 = 0$
E) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$

Question 28

If $\mathrm { X } _ { 1 } , \mathrm { X } _ { 2 },$ and $X _ { 3 }$ are solutions of the third order system $\mathbf { X } ^ { \prime } = A \mathbf { X }$ and $\mathrm { X } _ { p }$ is a particular solution of $\mathrm { X } ^ { \prime } = A \mathrm { X } + f ( t )$ , then the general solution of $\mathrm { X } ^ { \prime } = A \mathrm { X } + f ( t )$ is

A) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + c _ { 3 } \mathbf { X } _ { 3 }$
B) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + c _ { 3 } \mathbf { X } _ { 3 } + c _ { 4 } \mathbf { X } _ { p }$
C) $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + c _ { 3 } \mathbf { X } _ { 3 } + \mathbf { X } _ { p }$
D) $\mathrm { X } _ { 1 } + \mathrm { X } _ { 2 } + \mathrm { X } _ { 3 } + \mathrm { X } _ { p }$
E) $\mathrm { X } _ { 1 } + \mathrm { X } _ { 2 } + \mathrm { X } _ { 3 } + c _ { 4 } \mathrm { X } _ { p }$

Question 29

A particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 1 \\t\end{array} \right)$ is

A) $\left( \begin{array} { l } t ^ { 2 } \\0\end{array} \right)$
B) $\left( \begin{array} { l } 0 \\t\end{array} \right)$
C) $\left( \begin{array} { l } 1 \\t\end{array} \right)$
D) $\left( \begin{array} { l } t \\0\end{array} \right)$
E) $\left( \begin{array} { l } t \\1\end{array} \right)$

Question 30

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 1 \\1 & - 1\end{array} \right)$ is

A) $\lambda ^ { 2 } + 4 \lambda - 4 = 0$
B) $\lambda ^ { 2 } + 4 \lambda + 4 = 0$
C) $\lambda ^ { 2 } - 2 \lambda = 0$
D) $\lambda ^ { 2 } + 2 \lambda = 0$
E) $\lambda ^ { 2 } - 2 \lambda - 4 = 0$

Question 31

The characteristic equation of $A = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right)$ is

A) $\lambda ^ { 2 } + 3 \lambda - 2 = 0$
B) $\lambda ^ { 2 } + 5 \lambda + 2 = 0$
C) $\lambda ^ { 2 } - 5 \lambda + 2 = 0$
D) $\lambda ^ { 2 } + 3 \lambda + 2 = 0$
E) $\lambda ^ { 2 } - 3 \lambda + 2 = 0$

Question 32

The eigenvalues of the matrix A of the previous problem are

A) $- 2 , - 2,3$
B) $2,2,3$
C) $1 \pm i , 3$
D) $- 1 \pm i , 3$
E) $1 \pm \sqrt { 3 } i , 3$

Question 33

The eigenvalues of $A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)$ is

A) $0,4$
B) $0 , - 4$
C) $2,2$
D) $2 \pm 2 i$
E) $- 2 \pm 2 i$

Question 34

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

A) $1 \pm \sqrt { 3 } i$
B) $- 1 \pm \sqrt { 3 } i$
C) $0,3$
D) $0,1$
E) $0 , - 1$

Question 35

The Wronskian of the vector functions $X _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }$ and $X _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }$ is

A) $8 e ^ { 4 t }$
B) $2 e ^ { 4 t }$
C) $- 8 e ^ { 4 t }$
D) $- 2 e ^ { 4 t }$
E) $15 e ^ { 4 t }$

Question 36

Let $A = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right)$ . Then $e ^ { A t } =$

A) $I \cos t + A \sin t$
B) $I \cos t - A \sin t$
C) $l e ^ { t } + A e ^ { - t }$
D) $l e ^ { - t } + A e ^ { - t }$
E) $I \cosh t + A \sinh t$

Question 37

The Wronskian of the vector functions $\mathbf { X } _ { 1 } = \left( \begin{array} { l } 2 \\1 \\0\end{array} \right) e ^ { t } , \mathbf { X } _ { 2 } = \left( \begin{array} { l } 1 \\3 \\5\end{array} \right) e ^ { 2 t }$ and $X _ { 3 } = \left( \begin{array} { c } - 1 \\5 \\0\end{array} \right) e ^ { 3 t }$ is

A) $55 e ^ { 6 t }$
B) $- 55 e ^ { 6 t }$
C) $45 e ^ { 6 t }$
D) $- 45 e ^ { - 6 t }$
E) $45 e ^ { - 6 t }$

Question 38

Let $A = \left( \begin{array} { l l l } 0 & 1 & 2 \\0 & 0 & 4 \\0 & 0 & 0\end{array} \right)$ . Then $e ^ { A t } =$

A) $\left( \begin{array} { l l l } 1 & t & 2 t \\0 & 1 & 4 t \\0 & 0 & 1\end{array} \right)$
B) $\left( \begin{array} { c c c } 1 & - t & 2 t + 2 t ^ { 2 } \\0 & 1 & - 4 t \\0 & 0 & 1\end{array} \right)$
C) $\left( \begin{array} { c c c } 1 & t & 2 t + 2 t ^ { 2 } \\0 & 1 & 4 t \\0 & 0 & 1\end{array} \right)$
D) $\left( \begin{array} { c c c } 1 & t & 2 t + 4 t ^ { 2 } \\0 & 1 & 4 t \\0 & 0 & 1\end{array} \right)$
E) none of the above

Question 39

The characteristic equation of $A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)$ is

A) $\lambda ^ { 2 } - 4 \lambda = 0$
B) $\lambda ^ { 2 } + 4 \lambda = 0$
C) $\lambda ^ { 2 } + 4 \lambda - 8 = 0$
D) $\lambda ^ { 2 } - 4 \lambda + 8 = 0$
E) $\lambda ^ { 2 } - 4 \lambda - 8 = 0$

Question 40

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } - 3 & - 1 \\1 & - 1\end{array} \right) X$ is

A) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left[ \left( \begin{array} { c } - 1 \\1\end{array} \right) t e ^ { - 2 t } + \left( \begin{array} { l } 1 \\0\end{array} \right) e ^ { - 2 t } \right]$
B) $\mathbf { X } = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left( \begin{array} { c } - 1 \\1\end{array} \right) t e ^ { - 2 t } + c _ { 3 } \left( \begin{array} { l } 1 \\0\end{array} \right) e ^ { - 2 t }$
C) $X = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) t e ^ { - 2 t }$
D) $X = c _ { 1 } \left( \begin{array} { c } - 1 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left( \begin{array} { l } 1 \\2\end{array} \right) e ^ { - t }$
E) none of the above