Deck 17: Mathematical Problems and Solutions

Question 1

The solution of $y ^ { \prime \prime } - 6 y ^ { \prime } + 8 y = 0$ is

A) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 4 x }$
B) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } x e ^ { 4 x }$
C) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } x e ^ { - 4 x }$
D) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 4 x }$
E) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { - 4 x }$

Answer 1

$y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 4 x }$

Question 2

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

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) $$X = c _ { 1 } \left( \begin{array} { c } 1 \\- 2\end{array} \right) e ^ { 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { 3 t }$$

Answer 2

$$X = c _ { 1 } \left( \begin{array} { c } 2 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { 3 t }$$

Question 3

Using the convolution theorem, we find that $\mathcal { L } ^ { - 1 } \left\{ 1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right) \right\} =$

A) $\left( e ^ { - t } + \sin t - \cos t \right) / 2$
B) $\left( e ^ { t } + \sin t - \cos t \right) / 2$
C) $\left( e ^ { - t } + \sin t + \cos t \right) / 2$
D) $\left( e ^ { t } - \sin t - \cos t \right) / 2$
E) $\left( e ^ { - t } - \sin t - \cos t \right) / 2$

Answer 3

$\left( e ^ { - t } + \sin t - \cos t \right) / 2$

Question 4

Using power series methods, the solution of $2 x y ^ { \prime \prime } + y ^ { \prime } + 2 y = 0$ is

A) $\begin{array} { l } y = c _ { 0 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\c _ { 1 } x ^ { 1 / 2 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) )\end{array}$
B) $\begin{array} { l } y = c _ { 0 } \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$
C) $\begin{array} { l } y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\c _ { 1 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$
D) $\begin{array} { l } y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$
E) $\begin{array} { l } y = \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$

Question 5

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

A) $y = c _ { 1 } + c _ { 2 } e ^ { - 2 x } + x ^ { 2 } / 4 - x / 4 + e ^ { x } / 3$
B) $y = c _ { 1 } + c _ { 2 } e ^ { - 2 x } + x ^ { 2 } / 4 + x / 4 - e ^ { x } / 3$
C) $y = c _ { 1 } + c _ { 2 } e ^ { - 2 x } + x ^ { 2 } / 4 + x / 4 + e ^ { x } / 3$
D) $y = c _ { 1 } + c _ { 2 } e ^ { - 2 x } - x ^ { 2 } / 4 - x / 4 - e ^ { x } / 3$
E) $y = c _ { 1 } + c _ { 2 } e ^ { - 2 x } - x ^ { 2 } / 4 - x / 4 + e ^ { x } / 3$

Question 6

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

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) $y = - 1 / ( x - \ln x + c )$

Question 7

In the previous problem, the solution for the temperature is

A) $T ( t ) = 70 - 38 e ^ { - .930 t }$
B) $T ( t ) = 70 - 38 e ^ { .930 t }$
C) $T ( t ) = 55 - 32 e ^ { - .930 t }$
D) $T ( t ) = 55 - 32 e ^ { .930 t }$
E) $T ( t ) = 55 e ^ { - .930 t }$

Question 8

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

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

Question 9

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

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 10

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

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 11

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

A) $y = \sin t + \sin ( t - \pi / 2 ) u ( t - \pi / 2 )$
B) $y = \sin t - \cos ( t - \pi / 2 ) \boldsymbol { u } ( t - \pi / 2 )$
C) $y = \cos t + \sin ( t - \pi / 2 ) u ( t - \pi / 2 )$
D) $y = \cos t + \cos ( t - \pi / 2 ) \boldsymbol { u } ( t - \pi / 2 )$
E) $y = \cos t - \sin ( t - \pi / 2 ) u ( t - \pi / 2 )$

Question 12

The solution of $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = \cos x$ is

A) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( 5 \sin x + 3 \cos x ) / 34$
B) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \sin x + 3 \cos x ) / 34$
C) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \cos x - 3 \sin x ) / 34$
D) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( 5 \cos x + 3 \sin x ) / 34$
E) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \cos x + 3 \sin x ) / 34$

Question 13

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

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

Question 14

A 4-pound weight is hung on a spring and stretches it 1 foot. The mass spring system is then put into motion in a medium offering a damping force numerically equal to the velocity. If the mass is pulled down 6 inches from equilibrium and released, the initial value problem describing the position, $x ( t )$ , of the mass at time t is

A) $x ^ { \prime \prime } - 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 6 , x ^ { \prime } ( 0 ) = 0$
B) $x ^ { \prime \prime } + 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 6 , x ^ { \prime } ( 0 ) = 0$
C) $x ^ { \prime \prime } - 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$
D) $x ^ { \prime \prime } + 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$
E) $x ^ { \prime \prime } + 32 x = 8 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$

Question 15

The solution of $$X ^ { \prime } = \left( \begin{array} { c c c } 4 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) X$$ are

A) $$X = c _ { 1 } \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 4 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
B) $$X = c _ { 1 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 4 t } + c _ { 2 } \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
C) $$X = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 4 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
D) $$X = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 4 t } + c _ { 2 } \left( \begin{array} { l } 0 \\1 \\1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { l } 0 \\1 \\1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
E) $$X = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 4 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$

Question 16

The solution of $y ^ { \prime \prime } - 4 y ^ { \prime } + 20 y = 0$ is

A) $y = c _ { 1 } e ^ { - 2 x } \cos ( 4 x ) + c _ { 2 } e ^ { - 2 x } \sin ( 4 x )$
B) $y = c _ { 1 } e ^ { - 2 x } \cos ( 4 x ) + c _ { 2 } e ^ { 2 x } \sin ( 4 x )$
C) $y = c _ { 1 } e ^ { 2 x } \cos ( 4 x ) + c _ { 2 } e ^ { 2 x } \sin ( 4 x )$
D) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 4 x }$
E) $y = c _ { 1 } \cos ( 4 x ) + c _ { 2 } \sin ( 4 x )$

Question 17

Using power series methods, the solution of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ is

A) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / n ! \right]$
B) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n + 1 ) ) \right]$
C) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
D) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
E) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$

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)

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

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

A) $y = c _ { 1 } \cos x + c _ { 2 } \sin x + \cos x \ln | \sec x + \tan x |$
B) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \sec x + \tan x |$
C) $y = c _ { 1 } \cos x + c _ { 2 } \sin x + \cos x \ln | \sec x |$
D) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \tan x |$
E) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \sec x - \tan x |$

Question 20

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

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 21

In the previous two problems, the error in the improved Euler method at $x = 0.1$ is

A) $0.00467$
B) 0.000165
C) 0.870
D) 0.895
E) 0.0897

Question 22

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

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 23

The solutions of the eigenvalue problem and the other problem from the previous problem are

A) $\lambda = n \pi , X = \cos ( n \pi x ) , Y = \sinh ( n \pi y ) , n = 1,2,3 , \ldots$
B) $\lambda = n \pi , X = \sin ( n \pi x ) , Y = \sinh ( n \pi y ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } \pi ^ { 2 } , X = \cos ( n \pi x ) , Y = \sinh ( n \pi y ) , n = 1,2,3 , \ldots$
D) $\lambda = n ^ { 2 } \pi ^ { 2 } , X = \sin ( n \pi x ) , Y = \sinh ( n \pi y ) , n = 1,2,3 , \ldots$
E) $\lambda = n ^ { 2 } \pi ^ { 2 } , X = \cos ( n \pi x ) , Y = \sinh ( n \pi y ) , n = 0,1,2 , \ldots , ( Y = y \text { if } n = 0 )$

Question 24

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

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

Question 25

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

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 26

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

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 27

Consider the non-linear system $x ^ { \prime } = 1 - 2 x y , y ^ { \prime } = 2 x y - y$ . The linearized system about the one critical point, $( 1 / 2,1 ) \text {, is } X ^ { t } = A X \text {, where } A =$

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

Question 28

Consider the heat problem $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , 0 < x < \infty , t > 0 , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$ . Apply a Fourier sine transform. The resulting problem for $U ( \alpha , t ) = \mathcal { F } _ { s } \{ u ( x , t ) \}$ is

A) $U _ { t } = - k \alpha U + k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$
B) $U _ { t } = - k \alpha ^ { 2 } U - k \alpha t _ { 0 } , U ( \alpha , 0 ) = 0$
C) $U _ { t } = - k \alpha ^ { 2 } U + k \alpha t _ { 0 } , U ( \alpha , 0 ) = 0$
D) $U _ { t } = k \alpha ^ { 2 } U + k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$
E) $U _ { t } = k \alpha ^ { 2 } U - k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$

Question 29

Let $A = \left( \begin{array} { c c } - 4 & - 9 \\4 & 8\end{array} \right)$ , and consider the system $X ^ { \prime } = A X$ . The critical point $( 0,0 )$ of the system is a

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

Question 30

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

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 31

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

A) $y _ { 1 } = 0.1015$
B) $y _ { 1 } = 0.115$
C) $y _ { 1 } = 0.105$
D) $y _ { 1 } = 0.10005$
E) $y _ { 1 } = 0.1005$

Question 32

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

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 33

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

A) $y = \tan x$
B) $y = \sec x$
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 34

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

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

Question 35

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

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) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$

Question 36

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

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 37

Let $A = \left( \begin{array} { c c } - 1 & - 4 \\1 & - 1\end{array} \right)$ , and consider the system $X ^ { \prime } = A X$ . The critical point $( 0,0 )$ of the system is a spiral point. The origin is

A) unstable, and the solutions recede from the origin clockwise as $t \rightarrow \infty$ .
B) unstable, and the solutions recede from the origin counter-clockwise as $t \rightarrow \infty$ .
C) stable, and the solutions approach the origin clockwise as $t \rightarrow \infty$ .
D) stable, and the solutions approach the origin counter-clockwise as $t \rightarrow \infty$ .
E) none of the above

Question 38

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

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 39

The solutions for $\lambda , R \text { and } \Theta$ from the previous problem are

A) $\lambda = n ^ { 2 } , R = r ^ { n } , \Theta = \cos ( n \theta ) , n = 1,2,3 , \ldots$
B) $\lambda = n ^ { 2 } , R = r ^ { n } , \Theta = \sin ( n \theta ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } , R = r ^ { n } , \Theta = \sin ( n \theta ) , n = 0,1,2 , \ldots$
D) $\lambda = n , R = r ^ { n } , \Theta = \sin ( n \theta ) , n = 1,2,3 , \ldots$
E) $\lambda = n , R = r ^ { n } , \Theta = \cos ( n \theta ) , n = 1,2,3 , \ldots$

Question 40

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

A) 0.099589
B) 0.100334589
C) 0.10034589
D) 0.10334589
E) 0.1034589

Question 41

In the previous two problems, the solution for u along the line $t = 0.5$ at the mesh points is Select all that apply.

A) $u _ { 11 } = 10 / 3$
B) $u _ { 11 } = 20 / 9$
C) $u _ { 11 } = 20 / 3$
D) $u _ { 21 } = 32 / 3$
E) $u _ { 21 } = 13 / 3$

Question 42

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

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 43

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

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

Question 44

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

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

Question 45

The eigenvalue-eigenvector pairs for the matrix $$A = \left( \begin{array} { c c c } 4 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }$$ are Select all that apply.

A) $$4 , \left( \begin{array} { l } 1 \\0 \\0\end{array} \right)$$
B) $$2 , \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right)$$
C) $$2 , \left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right)$$
D) $$2 , \left( \begin{array} { l } 0 \\1 \\1\end{array} \right)$$
E) $$2 , \left( \begin{array} { l } 0 \\1 \\0\end{array} \right)$$

Question 46

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

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 47

The solutions of a regular Sturm-Liouville problem $\left( \left( r y ^ { \prime } \right) ^ { \prime } + ( \lambda p + q ) y = 0 , y ( a ) = 0 , y ( b ) = 0 \right)$ have which of the following properties?

A) There exists an infinite number of real eigenvalues.
B) The eigenvalues are orthogonal on $[ a , b ]$ .
C) For each eigenvalue, there is only one eigenfunction (except for non-zero constant multiples).
D) Eigenfunctions corresponding to different eigenvalues are linearly independent.
E) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function $r ( x )$ on the interval $[ a , b ]$ .

Question 48

Consider the heat problem $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 1 , t ) = 3 , u ( x , 0 ) = 3 x ^ { 2 }$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 3$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 2$ . The resulting equation is

A) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
B) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
C) $c [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
D) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
E) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$