Question 1

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

Accepted Answer

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

Question 2

The solution of $$X ^ { \prime } = \left( \begin{array} { c c c } 
3 & 0 & 0 \
0 & 3 & 1 \
0 & - 1 & 1
\end{array} \right) \mathrm { X }$$ is

Accepted Answer

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

Question 3

The differential equation $y ^ { \prime } + y = x ^ { 2 }$ is Select all that apply.

Accepted Answer

A)  linear 
B)  separable 
C)  exact 
D)  non-linear 
E)  Bernoulli 
A)  linear 
B)  separable 
C)  exact 
D)  non-linear 
E)  Bernoulli

Question 4

Using the improved Euler method with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

Accepted Answer

A)  $y _ { 1 } = 0.095$ 
B)  $y _ { 1 } = 0.995$ 
C)  $y _ { 1 } = 0.95$ 
D)  $y _ { 1 } = 0.00995$ 
E)  $y _ { 1 } = 0.0995$ 
A)  $y _ { 1 } = 0.095$ 
B)  $y _ { 1 } = 0.995$ 
C)  $y _ { 1 } = 0.95$ 
D)  $y _ { 1 } = 0.00995$ 
E)  $y _ { 1 } = 0.0995$

Question 5

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

Accepted Answer

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

Question 6

In the previous problem, the exact solution of the initial value problem is

Accepted Answer

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

Question 7

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

Accepted Answer

A)  $y = e ^ { t } + t e ^ { t } + t ^ { 2 } e ^ { t } / 2$ 
B)  $y = e ^ { - t } + t e ^ { - t } + t ^ { 2 } e ^ { - t } / 2$ 
C)  $y = e ^ { t } - t e ^ { t } + t ^ { 2 } e ^ { t } / 2$ 
D)  $y = e ^ { - t } - t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$ 
E)  $y = e ^ { - t } + t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$ 
A)  $y = e ^ { t } + t e ^ { t } + t ^ { 2 } e ^ { t } / 2$ 
B)  $y = e ^ { - t } + t e ^ { - t } + t ^ { 2 } e ^ { - t } / 2$ 
C)  $y = e ^ { t } - t e ^ { t } + t ^ { 2 } e ^ { t } / 2$ 
D)  $y = e ^ { - t } - t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$ 
E)  $y = e ^ { - t } + t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$

Question 8

In the previous two problems, the solution for the temperature is

Accepted Answer

A)  $T ( t ) = 20 - 20 e ^ { - 230 t }$ 
B)  $T ( t ) = 20 - 20 e ^ { 230 t }$ 
C)  $T ( t ) = 18 - 18 e ^ { - 230 t }$ 
D)  $T ( t ) = 18 - 18 e ^ { 230 t }$ 
E)  $T ( t ) = 18 e ^ { - 230 t }$ 
A)  $T ( t ) = 20 - 20 e ^ { - 230 t }$ 
B)  $T ( t ) = 20 - 20 e ^ { 230 t }$ 
C)  $T ( t ) = 18 - 18 e ^ { - 230 t }$ 
D)  $T ( t ) = 18 - 18 e ^ { 230 t }$ 
E)  $T ( t ) = 18 e ^ { - 230 t }$

Question 9

The solution of the previous problem is

Accepted Answer

A)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$ 
B)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 12 E I )$ 
C)  $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 24$ 
D)  $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 12$ 
E)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$ 
A)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$ 
B)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 12 E I )$ 
C)  $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 24$ 
D)  $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 12$ 
E)  $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$

Question 10

The auxiliary equation of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

Accepted Answer

A)  $m ^ { 2 } - 5 m - 6 = 0$ 
B)  $m ^ { 2 } - 5 m + 6 = 0$ 
C)  $m ^ { 2 } - 5 m + 6 = 1$ 
D)  $m ^ { 2 } - 5 m + 6$ 
E)  $m ^ { 2 } - 5 m - 6$ 
A)  $m ^ { 2 } - 5 m - 6 = 0$ 
B)  $m ^ { 2 } - 5 m + 6 = 0$ 
C)  $m ^ { 2 } - 5 m + 6 = 1$ 
D)  $m ^ { 2 } - 5 m + 6$ 
E)  $m ^ { 2 } - 5 m - 6$

Question 11

The solution of $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } = 0$ is

Accepted Answer

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

Question 12

The solution of the system in the previous problem is

Accepted Answer

A)  $y _ { 1 } = 1.228 , y _ { 2 } = 1.482 , y _ { 3 } = 1.753$ 
B)  $y _ { 1 } = 1.228 , y _ { 2 } = 1.646 , y _ { 3 } = 1.753$ 
C)  $y _ { 1 } = 1.126 , y _ { 2 } = 1.646 , y _ { 3 } = 2.903$ 
D)  $y _ { 1 } = 1.126 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$ 
E)  $y _ { 1 } = 1.016 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$ 
A)  $y _ { 1 } = 1.228 , y _ { 2 } = 1.482 , y _ { 3 } = 1.753$ 
B)  $y _ { 1 } = 1.228 , y _ { 2 } = 1.646 , y _ { 3 } = 1.753$ 
C)  $y _ { 1 } = 1.126 , y _ { 2 } = 1.646 , y _ { 3 } = 2.903$ 
D)  $y _ { 1 } = 1.126 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$ 
E)  $y _ { 1 } = 1.016 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$

Question 13

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + y = \delta ( t - \pi ) , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

Accepted Answer

A)  $y = \sin t + \sin ( t - \pi ) \boldsymbol { u } ( t - \pi )$ 
B)  $y = \sin t - \cos ( t - \pi ) u ( t - \pi )$ 
C)  $y = \cos t + \sin ( t - \pi ) u ( t - \pi )$ 
D)  $y = \cos t + \cos ( t - \pi ) u ( t - \pi )$ 
E)  $y = \cos t - \sin ( t - \pi ) u ( t - \pi )$ 
A)  $y = \sin t + \sin ( t - \pi ) \boldsymbol { u } ( t - \pi )$ 
B)  $y = \sin t - \cos ( t - \pi ) u ( t - \pi )$ 
C)  $y = \cos t + \sin ( t - \pi ) u ( t - \pi )$ 
D)  $y = \cos t + \cos ( t - \pi ) u ( t - \pi )$ 
E)  $y = \cos t - \sin ( t - \pi ) u ( t - \pi )$

Question 14

The solution of $x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0$ is

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

A uniform beam of length 10 has a concentrated load $w _ { 0 }$ at $x = 5$ . It is embedded at both ends. The boundary value problem for the deflections, $y ( x )$ , for this system is

Accepted Answer

A)  $y ^ { \prime \prime \prime \prime } = E L w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
B)  $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
C)  $E l y ^ { \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
D)  $E L y ^ { \prime \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
E)  $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
A)  $y ^ { \prime \prime \prime \prime } = E L w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
B)  $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
C)  $E l y ^ { \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
D)  $E L y ^ { \prime \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$ 
E)  $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$

Question 17

Let $A = \left( \begin{array} { c c } 
2 & 0 \
0 & - 3
\end{array} \right)$ . Then $e ^ { A t } =$

Accepted Answer

A)  $I + A t$ 
B)  $I + A t + A ^ { 2 } t ^ { 2 } / 2$ 
C)  $A t + A ^ { 2 } t ^ { 2 } / 2$ 
D)  $\left( \begin{array} { c c } 
e ^ { 2 t } & 0 \
0 & e ^ { - 3 t }
\end{array} \right)$ 
E)  $\left( \begin{array} { c c } 
e ^ { 2 t - 1 } - 1 & 0 \
0 & e ^ { - 3 t } - 1
\end{array} \right)$ 
A)  $I + A t$ 
B)  $I + A t + A ^ { 2 } t ^ { 2 } / 2$ 
C)  $A t + A ^ { 2 } t ^ { 2 } / 2$ 
D)  $\left( \begin{array} { c c } 
e ^ { 2 t } & 0 \
0 & e ^ { - 3 t }
\end{array} \right)$ 
E)  $\left( \begin{array} { c c } 
e ^ { 2 t - 1 } - 1 & 0 \
0 & e ^ { - 3 t } - 1
\end{array} \right)$

Question 18

The differential equation $y ^ { \prime } = x ^ { 2 } y ^ { 2 }$ is Select all that apply.

Accepted Answer

A)  linear 
B)  separable 
C)  exact 
D)  non-linear 
E)  Bernoulli 
A)  linear 
B)  separable 
C)  exact 
D)  non-linear 
E)  Bernoulli

Question 19

Using the classical Runge-Kutta method of order 4 with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

Accepted Answer

A)  0.099588 
B)  0.099668 
C)  0.099688 
D)  0.099768 
E)  0.099788 
A)  0.099588 
B)  0.099668 
C)  0.099688 
D)  0.099768 
E)  0.099788

Question 20

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ is

Accepted Answer

A)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
B)  $\lambda = n \pi / 2 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \sin ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = ( 2 n - 1 ) \pi / 2 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$ 
E)  $\lambda = ( 2 n - 1 ) ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$ 
A)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
B)  $\lambda = n \pi / 2 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \sin ( n \pi x / 2 ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = ( 2 n - 1 ) \pi / 2 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$ 
E)  $\lambda = ( 2 n - 1 ) ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$

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

The solution of $X ^ { \prime } = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }$ is

The differential equation $y ^ { \prime } + y = x ^ { 2 }$ is Select all that apply.

Using the improved Euler method with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

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

In the previous problem, the exact solution of the initial value problem is

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

In the previous two problems, the solution for the temperature is

The solution of the previous problem is

The auxiliary equation of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

The solution of $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } = 0$ is

The solution of the system in the previous problem is

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + y = \delta ( t - \pi ) , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

The solution of $x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0$ is

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

A uniform beam of length 10 has a concentrated load $w _ { 0 }$ at $x = 5$ . It is embedded at both ends. The boundary value problem for the deflections, $y ( x )$ , for this system is

Let $A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)$ . Then $e ^ { A t } =$

The differential equation $y ^ { \prime } = x ^ { 2 } y ^ { 2 }$ is Select all that apply.

Using the classical Runge-Kutta method of order 4 with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ 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

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

Mathematical Problems and Solutions

Filters

Exam 16: Mathematics Problems: Differential Equations and Linear Algebra

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

The solution of X′=(3000310−11)XX ^ { \prime } = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }X′=​300​03−1​011​​X is

The differential equation y′+y=x2y ^ { \prime } + y = x ^ { 2 }y′+y=x2 is Select all that apply.

Using the improved Euler method with a step size of h=0.1h = 0.1h=0.1 , the solution of y′=1−y2,y(0)=0y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0y′=1−y2,y(0)=0 at x=0.1x = 0.1x=0.1 is

The solution of X′=(1−21−1)X\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X }X′=(11​−2−1​)X is

In the previous problem, the exact solution of the initial value problem is

Using Laplace transform methods, the solution of y′′+2y′+y=e−t,y(0)=1,y′(0)=0y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0y′′+2y′+y=e−t,y(0)=1,y′(0)=0 is

In the previous two problems, the solution for the temperature is

The solution of the previous problem is

The auxiliary equation of y′′−5y′+6y=0y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0y′′−5y′+6y=0 is

The solution of x2y′′+xy′=0x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } = 0x2y′′+xy′=0 is

The solution of the system in the previous problem is

Using Laplace transform methods, the solution of y′′+y=δ(t−π),y(0)=1,y′(0)=0y ^ { \prime \prime } + y = \delta ( t - \pi ) , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0y′′+y=δ(t−π),y(0)=1,y′(0)=0 is

The solution of x2ydx+(x3/3+y)dy=0x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0x2ydx+(x3/3+y)dy=0 is

The eigenvalues of the matrix A=(1−21−1)A = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right)A=(11​−2−1​) are

A uniform beam of length 10 has a concentrated load w0w _ { 0 }w0​ at x=5x = 5x=5 . It is embedded at both ends. The boundary value problem for the deflections, y(x)y ( x )y(x) , for this system is

Let A=(200−3)A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)A=(20​0−3​) . Then eAt=e ^ { A t } =eAt=

The differential equation y′=x2y2y ^ { \prime } = x ^ { 2 } y ^ { 2 }y′=x2y2 is Select all that apply.

Using the classical Runge-Kutta method of order 4 with a step size of h=0.1h = 0.1h=0.1 , the solution of y′=1−y2,y(0)=0y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0y′=1−y2,y(0)=0 at x=0.1x = 0.1x=0.1 is

The solution of the eigenvalue problem y′′+λy=0,y′(0)=0,y(1)=0y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0y′′+λy=0,y′(0)=0,y(1)=0 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

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

Mathematical Problems and Solutions

Filters

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

The solution of $X ^ { \prime } = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }$ is

The differential equation $y ^ { \prime } + y = x ^ { 2 }$ is Select all that apply.

Using the improved Euler method with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

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

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

The auxiliary equation of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

The solution of $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } = 0$ is

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + y = \delta ( t - \pi ) , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

The solution of $x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0$ is

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

A uniform beam of length 10 has a concentrated load $w _ { 0 }$ at $x = 5$ . It is embedded at both ends. The boundary value problem for the deflections, $y ( x )$ , for this system is

Let $A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)$ . Then $e ^ { A t } =$

The differential equation $y ^ { \prime } = x ^ { 2 } y ^ { 2 }$ is Select all that apply.

Using the classical Runge-Kutta method of order 4 with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ is