Question 1

Is the value of $\lambda$ in the previous problem such that the scheme is stable?

Accepted Answer

A)  yes 
B)  no 
C)  It is right on the borderline. 
D)  It cannot be determined from the available data. 
A)  yes 
B)  no 
C)  It is right on the borderline. 
D)  It cannot be determined from the available data. B

Question 2

The solution of $x y ^ { \prime } = ( x - 1 ) y ^ { 2 }$ is

Accepted Answer

A)  $y = 1 / ( x + \ln x + c )$ 
B)  $y = 1 / ( x - \ln x + c )$ 
C)  $y = - c / ( x + \ln x )$ 
D)  $y = - c / ( x - \ln x )$ 
E)  $y = - 1 / ( x - \ln x + c )$ 
A)  $y = 1 / ( x + \ln x + c )$ 
B)  $y = 1 / ( x - \ln x + c )$ 
C)  $y = - c / ( x + \ln x )$ 
D)  $y = - c / ( x - \ln x )$ 
E)  $y = - 1 / ( x - \ln x + c )$ E

Question 3

In the previous two problems, the solution for $u ( x , y )$ is

Accepted Answer

A)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi x ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
B)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \sinh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
C)  $u = c _ { 0 } y + \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi y )$, where $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) d x / 4$ and
$$c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$$ 
D)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
E)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
A)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi x ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
B)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \sinh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
C)  $u = c _ { 0 } y + \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi y )$, where $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) d x / 4$ and
$$c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$$ 
D)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ 
E)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$ C

Question 4

In the previous problem, using the notation $u _ { i j } = u ( x , t )$ , and letting $c = 1 , \lambda = c k / h ^ { 2 }$ , the equation becomes

Accepted Answer

A)  $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
B)  $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
C)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
D)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
E)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
A)  $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
B)  $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
C)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
D)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
E)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$

Question 5

Consider the problem $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0$ with boundary conditions $u ( r , 0 ) = 0$ , $u ( r , \pi ) = 0 , u ( 1 , \theta ) = f ( \theta )$ . Separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems for $R \text { and } \Theta$ are

Accepted Answer

A)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
B)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
C)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
D)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) \text { is bounded, } \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
E)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
A)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
B)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
C)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
D)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , R ( 0 ) \text { is bounded, } \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
E)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$

Question 6

Using Laplace transform methods, the solution of $y ^ { \prime } + y = 2 \sin t , y ( 0 ) = 1$ is (Hint: the previous problem might be useful.)

Accepted Answer

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

Question 7

In the previous problem, the solution for $U ( \alpha , t )$ is

Accepted Answer

A)  $U = u _ { 0 } \left( 1 + e ^ { - k \alpha ^ { 2 } t } \right) / \alpha$ 
B)  $U = u _ { 0 } \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) / \alpha$ 
C)  $U = u _ { 0 } \left( 1 - e ^ { k \alpha ^ { 2 } t } \right) / \alpha$ 
D)  $U = u _ { 0 } \left( 1 - e ^ { k \alpha ^ { 2 } t } \right)$ 
E)  $U = u _ { 0 } \left( 1 + e ^ { - k \alpha ^ { 2 } t } \right)$ 
A)  $U = u _ { 0 } \left( 1 + e ^ { - k \alpha ^ { 2 } t } \right) / \alpha$ 
B)  $U = u _ { 0 } \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) / \alpha$ 
C)  $U = u _ { 0 } \left( 1 - e ^ { k \alpha ^ { 2 } t } \right) / \alpha$ 
D)  $U = u _ { 0 } \left( 1 - e ^ { k \alpha ^ { 2 } t } \right)$ 
E)  $U = u _ { 0 } \left( 1 + e ^ { - k \alpha ^ { 2 } t } \right)$

Question 8

In the previous problem, for both the linearized system and the non-linear system, the critical point is a

Accepted Answer

A)  unstable node 
B)  stable node 
C)  saddle point 
D)  unstable spiral point 
E)  stable spiral point 
A)  unstable node 
B)  stable node 
C)  saddle point 
D)  unstable spiral point 
E)  stable spiral point

Question 9

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

Accepted Answer

A)  $$X _ { p } = \left( \begin{array} { c } 
t + 2 \
- t + 3
\end{array} \right)$$ 
B)  $$X _ { p } = \left( \begin{array} { c } 
t + 2 \
- t - 3
\end{array} \right)$$ 
C)  $$X _ { p } = \left( \begin{array} { l } 
- t + 2 \
- t - 3
\end{array} \right)$$ 
D)  $$\mathbf { X } _ { p } = \left( \begin{array} { c } 
- t + 2 \
t + 3
\end{array} \right)$$ 
E)  $$X _ { p } = \left( \begin{array} { l } 
- t - 2 \
- t + 3
\end{array} \right)$$ 
A)  $$X _ { p } = \left( \begin{array} { c } 
t + 2 \
- t + 3
\end{array} \right)$$ 
B)  $$X _ { p } = \left( \begin{array} { c } 
t + 2 \
- t - 3
\end{array} \right)$$ 
C)  $$X _ { p } = \left( \begin{array} { l } 
- t + 2 \
- t - 3
\end{array} \right)$$ 
D)  $$\mathbf { X } _ { p } = \left( \begin{array} { c } 
- t + 2 \
t + 3
\end{array} \right)$$ 
E)  $$X _ { p } = \left( \begin{array} { l } 
- t - 2 \
- t + 3
\end{array} \right)$$

Question 10

Consider Laplace's equation on a rectangle, $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$ with boundary conditions $u _ { x } ( 0 , y ) = 0 , u _ { x } ( 1 , y ) = 0 , u ( x , 0 ) = 0 , u ( x , 2 ) = f ( x )$ . When the variables are separated using $u ( x , y ) = X ( x ) Y ( y )$ , the resulting problems for $X$ and $Y$ are

Accepted Answer

A)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0$ 
B)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0$ 
C)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 2 ) = 0$ 
D)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$ 
E)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$ 
A)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0$ 
B)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0$ 
C)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 2 ) = 0$ 
D)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$ 
E)  $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$

Question 11

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

Accepted Answer

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

Question 12

The Fourier series of an even function can contain Select all that apply.

Accepted Answer

A)  sine terms 
B)  cosine terms 
C)  a constant term 
D)  more than one of the above 
E)  none of the above 
A)  sine terms 
B)  cosine terms 
C)  a constant term 
D)  more than one of the above 
E)  none of the above

Question 13

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

Accepted Answer

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

Question 14

In the previous two problems, the infinite series solution for $u ( r , \theta )$ is $u = \sum _ { n = 1 } ^ { \infty } c _ { n } r ^ { n } \Theta _ { n } ( \theta )$ , where $\Theta _ { n }$ is found in the previous problem, and

Accepted Answer

A)  $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$ 
B)  $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$ 
C)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$ 
D)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$ 
E)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / ( 2 \pi )$ 
A)  $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$ 
B)  $c _ { n } = 2 \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$ 
C)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \cos ( n \theta ) d \theta / \pi$ 
D)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / \pi$ 
E)  $c _ { n } = \int _ { 0 } ^ { \pi } f ( \theta ) \sin ( n \theta ) d \theta / ( 2 \pi )$

Question 15

The correct form of the particular solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }$ is

Accepted Answer

A)  $y _ { p } = A e ^ { - x }$ 
B)  $y _ { p } = A x e ^ { - x }$ 
C)  $y _ { p } = A x ^ { 2 } e ^ { - x }$ 
D)  $y _ { p } = A x ^ { 3 } e ^ { - x }$ 
E)  none of the above 
A)  $y _ { p } = A e ^ { - x }$ 
B)  $y _ { p } = A x e ^ { - x }$ 
C)  $y _ { p } = A x ^ { 2 } e ^ { - x }$ 
D)  $y _ { p } = A x ^ { 3 } e ^ { - x }$ 
E)  none of the above

Question 16

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 2 ) = 0$ is

Accepted Answer

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

Question 17

In the previous problem, the solution for the position, $x ( t )$ , is

Accepted Answer

A)  $x = e ^ { 4 t } ( \cos ( 4 t ) + \sin ( 4 t ) ) / 2$ 
B)  $x = e ^ { - 4 t } ( \cos ( 4 t ) + \sin ( 4 t ) ) / 2$ 
C)  $x = e ^ { - 4 t } ( \cos ( 8 t ) + \sin ( 8 t ) ) / 2$ 
D)  $x = e ^ { 4 t } ( \cos ( 8 t ) + \sin ( 8 t ) ) / 2$ 
E)  $x = 4 e ^ { 8 t } - 32 t e ^ { 8 t }$ 
A)  $x = e ^ { 4 t } ( \cos ( 4 t ) + \sin ( 4 t ) ) / 2$ 
B)  $x = e ^ { - 4 t } ( \cos ( 4 t ) + \sin ( 4 t ) ) / 2$ 
C)  $x = e ^ { - 4 t } ( \cos ( 8 t ) + \sin ( 8 t ) ) / 2$ 
D)  $x = e ^ { 4 t } ( \cos ( 8 t ) + \sin ( 8 t ) ) / 2$ 
E)  $x = 4 e ^ { 8 t } - 32 t e ^ { 8 t }$

Question 18

A frozen chicken at $32 ^ { \circ } \mathrm { F }$ is taken out of the freezer and placed on a table at $70 ^ { \circ } \mathrm { F }$ . One hour later the temperature of the chicken is $55 ^ { \circ } \mathrm { F }$ . The mathematical model for the temperature $T ( t )$ as a function of time $t$ is (assuming Newton 's law of warming)

Accepted Answer

A)  $\frac { d T } { d t } = k T , T ( 0 ) = 32 , T ( 1 ) = 55$ 
B)  $\frac { d T } { d t } = k ( T - 70 ) , T ( 0 ) = 32 , T ( 1 ) = 55$ 
C)  $\frac { d T } { d t } = ( T - 70 ) , T ( 0 ) = 32 , T ( 1 ) = 55$ 
D)  $\frac { d T } { d t } = T , T ( 0 ) = 32 , T ( 1 ) = 55$ 
E)  $\frac { d T } { d t } = k ( T - 55 ) , T ( 0 ) = 32 , T ( 1 ) = 55$ 
A)  $\frac { d T } { d t } = k T , T ( 0 ) = 32 , T ( 1 ) = 55$ 
B)  $\frac { d T } { d t } = k ( T - 70 ) , T ( 0 ) = 32 , T ( 1 ) = 55$ 
C)  $\frac { d T } { d t } = ( T - 70 ) , T ( 0 ) = 32 , T ( 1 ) = 55$ 
D)  $\frac { d T } { d t } = T , T ( 0 ) = 32 , T ( 1 ) = 55$ 
E)  $\frac { d T } { d t } = k ( T - 55 ) , T ( 0 ) = 32 , T ( 1 ) = 55$

Question 19

In the previous two problem, the solution for $u ( x , t )$ is

Accepted Answer

A)  $u = 2 u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha / \pi$ 
B)  $u = 2 u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha$ 
C)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha$ 
D)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha / \pi$ 
E)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \infty x ) / \alpha \right] d \alpha / ( 2 \pi )$ 
A)  $u = 2 u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha / \pi$ 
B)  $u = 2 u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha$ 
C)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha$ 
D)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \alpha x ) / \alpha \right] d \alpha / \pi$ 
E)  $u = u _ { 0 } \int _ { 0 } ^ { \infty } \left[ \left( 1 - e ^ { - k \alpha ^ { 2 } t } \right) \sin ( \infty x ) / \alpha \right] d \alpha / ( 2 \pi )$

Question 20

In the previous problem, the error in the classical Runge-Kutta method at $x = 0.1$ is (Hint: see the previous five problems.)

Accepted Answer

A)  0.00083 
B)  0.000083 
C)  0.000000083 
D)  0.0000083 
E)  0.00000083 
A)  0.00083 
B)  0.000083 
C)  0.000000083 
D)  0.0000083 
E)  0.00000083

Is the value of $\lambda$ in the previous problem such that the scheme is stable?

The solution of $x y ^ { \prime } = ( x - 1 ) y ^ { 2 }$ is

In the previous two problems, the solution for $u ( x , y )$ is

In the previous problem, using the notation $u _ { i j } = u ( x , t )$ , and letting $c = 1 , \lambda = c k / h ^ { 2 }$ , the equation becomes

Using Laplace transform methods, the solution of $y ^ { \prime } + y = 2 \sin t , y ( 0 ) = 1$ is (Hint: the previous problem might be useful.)

In the previous problem, the solution for $U ( \alpha , t )$ is

In the previous problem, for both the linearized system and the non-linear system, the critical point is a

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

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

The Fourier series of an even function can contain Select all that apply.

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

In the previous two problems, the infinite series solution for $u ( r , \theta )$ is $u = \sum _ { n = 1 } ^ { \infty } c _ { n } r ^ { n } \Theta _ { n } ( \theta )$ , where $\Theta _ { n }$ is found in the previous problem, and

The correct form of the particular solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 2 ) = 0$ is

In the previous problem, the solution for the position, $x ( t )$ , is

In the previous two problem, the solution for $u ( x , t )$ is

In the previous problem, the error in the classical Runge-Kutta method at $x = 0.1$ is (Hint: see the previous five problems.)

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

Mathematics Problems: Differential Equations and Linear Algebra

Filters

Exam 17: Mathematical Problems and Solutions

Is the value of λ\lambdaλ in the previous problem such that the scheme is stable?

The solution of xy′=(x−1)y2x y ^ { \prime } = ( x - 1 ) y ^ { 2 }xy′=(x−1)y2 is

In the previous two problems, the solution for u(x,y)u ( x , y )u(x,y) is

In the previous problem, using the notation uij=u(x,t)u _ { i j } = u ( x , t )uij​=u(x,t) , and letting c=1,λ=ck/h2c = 1 , \lambda = c k / h ^ { 2 }c=1,λ=ck/h2 , the equation becomes

Using Laplace transform methods, the solution of y′+y=2sin⁡t,y(0)=1y ^ { \prime } + y = 2 \sin t , y ( 0 ) = 1y′+y=2sint,y(0)=1 is (Hint: the previous problem might be useful.)

In the previous problem, the solution for U(α,t)U ( \alpha , t )U(α,t) is

In the previous problem, for both the linearized system and the non-linear system, the critical point is a

A particular solution of X′=(11−2−1)X+(2t)\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 1 & 1 \\- 2 & - 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 2 \\t\end{array} \right)X′=(1−2​1−1​)X+(2t​) is

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

The Fourier series of an even function can contain Select all that apply.

The solution of X′=(1−214)X\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 1 & - 2 \\1 & 4\end{array} \right) \mathbf { X }X′=(11​−24​)X is

In the previous two problems, the infinite series solution for u(r,θ)u ( r , \theta )u(r,θ) is u=∑n=1∞cnrnΘn(θ)u = \sum _ { n = 1 } ^ { \infty } c _ { n } r ^ { n } \Theta _ { n } ( \theta )u=∑n=1∞​cn​rnΘn​(θ) , where Θn\Theta _ { n }Θn​ is found in the previous problem, and

The correct form of the particular solution of y′′+2y′+y=e−xy ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }y′′+2y′+y=e−x is

The solution of the eigenvalue problem y′′+λy=0,y(0)=0,y(2)=0y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 2 ) = 0y′′+λy=0,y(0)=0,y(2)=0 is

In the previous problem, the solution for the position, x(t)x ( t )x(t) , is

In the previous two problem, the solution for u(x,t)u ( x , t )u(x,t) is

In the previous problem, the error in the classical Runge-Kutta method at x=0.1x = 0.1x=0.1 is (Hint: see the previous five problems.)

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

Mathematics Problems: Differential Equations and Linear Algebra

Filters

Is the value of $\lambda$ in the previous problem such that the scheme is stable?

The solution of $x y ^ { \prime } = ( x - 1 ) y ^ { 2 }$ is

In the previous two problems, the solution for $u ( x , y )$ is

In the previous problem, using the notation $u _ { i j } = u ( x , t )$ , and letting $c = 1 , \lambda = c k / h ^ { 2 }$ , the equation becomes

Using Laplace transform methods, the solution of $y ^ { \prime } + y = 2 \sin t , y ( 0 ) = 1$ is (Hint: the previous problem might be useful.)

In the previous problem, the solution for $U ( \alpha , t )$ is

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

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

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

In the previous two problems, the infinite series solution for $u ( r , \theta )$ is $u = \sum _ { n = 1 } ^ { \infty } c _ { n } r ^ { n } \Theta _ { n } ( \theta )$ , where $\Theta _ { n }$ is found in the previous problem, and

The correct form of the particular solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - x }$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 2 ) = 0$ is

In the previous problem, the solution for the position, $x ( t )$ , is

In the previous two problem, the solution for $u ( x , t )$ is

In the previous problem, the error in the classical Runge-Kutta method at $x = 0.1$ is (Hint: see the previous five problems.)