Question 1

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

Accepted Answer

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)  $I \cosh t + A \sinh t$ 
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)  $I \cosh t + A \sinh t$ A

Question 2

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

Accepted Answer

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)  $\lambda ^ { 2 } - 2 \lambda - 4 = 0$ 
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)  $\lambda ^ { 2 } - 2 \lambda - 4 = 0$ B

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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 
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 5

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

Accepted Answer

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)  $c _ { 1 } \mathbf { X } _ { 1 } + c _ { 2 } \mathbf { X } _ { 2 } + \mathbf { X } _ { p }$ 
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 6

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

Accepted Answer

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)  $\lambda ^ { 2 } - 4 \lambda - 8 = 0$ 
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 7

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

Accepted Answer

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 
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 8

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

Accepted Answer

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 
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

A)  $- 2 \pm \sqrt { 2 }$ 
B)  $2 \pm \sqrt { 2 }$ 
C)  $- 2 \pm 2 i$ 
D)  $2 \pm 2 i$ 
E)  $2 \pm 2 \sqrt { 3 }$ 
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 11

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

Accepted Answer

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)  $\lambda ^ { 2 } - 3 \lambda - 4 = 0$ 
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 12

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

Accepted Answer

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)  $\left( \begin{array} { c } 
e ^ { t } / 2 \
e ^ { t } / 2 - 1
\end{array} \right)$ 
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 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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 
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

Question 15

The eigenvalues of the matrix A of the previous problem are

Accepted Answer

A)  $- 2 , - 2,3$ 
B)  $2,2,3$ 
C)  $1 \pm i , 3$ 
D)  $- 1 \pm i , 3$ 
E)  $1 \pm \sqrt { 3 } i , 3$ 
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 16

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

Accepted Answer

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)  $\mathrm { X } _ { 1 } + \mathrm { X } _ { 2 } + \mathrm { X } _ { 3 } + c _ { 4 } \mathrm { X } _ { p }$ 
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 17

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

Accepted Answer

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 
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 18

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

Accepted Answer

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 
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 19

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

Accepted Answer

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 
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

Question 20

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

Accepted Answer

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

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

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

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

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

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

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

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

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

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

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

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

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

The eigenvalues of the matrix A of the previous problem are

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

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

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

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

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

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 8: Systems of Linear First-Order Differential Equations

Let A=(01−10)A = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right)A=(0−1​10​) . Then eAt=e ^ { A t } =eAt=

The characteristic equation of A=(−3−11−1)A = \left( \begin{array} { c c } - 3 & - 1 \\1 & - 1\end{array} \right)A=(−31​−1−1​) is

The Wronskian of the vector functions X1=(1−1)e−2tX _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }X1​=(1−1​)e−2t and X2=(35)e6tX _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }X2​=(35​)e6t is

The particular solution of X′=(2321)X+(e−t0)\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)X′=(22​31​)X+(e−t0​) is

The characteristic equation of A=(2−222)A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)A=(22​−22​) is

The characteristic equation of A=(1−12−1412−311−2)A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)A=​111​−1221​−14−3−2​​ is

Let A=(012004000)A = \left( \begin{array} { l l l } 0 & 1 & 2 \\0 & 0 & 4 \\0 & 0 & 0\end{array} \right)A=​000​100​240​​ . Then eAt=e ^ { A t } =eAt=

The eigenvalues of A=(−3−51−1)A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)A=(−31​−5−1​) is

The characteristic equation of A=(2211)A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)A=(21​21​) is

A particular solution of X′=(01−10)X+(1et)\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)X′=(0−1​10​)X+(1et​) is

The eigenvalues of A=(2−222)A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)A=(22​−22​) is

The solution of the system X′=(−3−11−1)XX ^ { \prime } = \left( \begin{array} { c c } - 3 & - 1 \\1 & - 1\end{array} \right) XX′=(−31​−1−1​)X is

The eigenvalues of the matrix A of the previous problem are

The general solution of X′=(2321)X+(t1)\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)X′=(22​31​)X+(t1​) is

The solution of the previous problem that satisfies the initial condition X(0)=(00)X ( 0 ) = \left( \begin{array} { l } 0 \\0\end{array} \right)X(0)=(00​) is

The solution of the system X′=(2211)X\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }X′=(21​21​)X is

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

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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