Question 1

In the previous three problems, the solution of the original problem is

Accepted Answer

A)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
B)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x$ 
C)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
D)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
A)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
B)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x$ 
C)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
D)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ D

Question 2

In the previous two problems, the product solutions are

Accepted Answer

A)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ 
B)  $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ 
C)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) / L }$ 
D)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ 
E)  $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ 
A)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ 
B)  $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ 
C)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) / L }$ 
D)  $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ 
E)  $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$ A

Question 3

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { x } - u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

Accepted Answer

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

Question 4

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u$ is

Accepted Answer

A)  first order, linear, homogeneous 
B)  first order, linear, non-homogeneous 
C)  second order, nonlinear 
D)  second order, linear, homogeneous 
E)  second order, linear, non-homogeneous 
A)  first order, linear, homogeneous 
B)  first order, linear, non-homogeneous 
C)  second order, nonlinear 
D)  second order, linear, homogeneous 
E)  second order, linear, non-homogeneous

Question 5

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

Accepted Answer

A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic 
A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic

Question 6

The solution of $y ^ { \prime \prime } - n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldots$ is

Accepted Answer

A)  $y = 0$ 
B)  $y = c \cdot \sin ( n x )$ 
C)  $y = c \cdot \cos ( n x )$ 
D)  $y = c \left( e ^ { n x } - e ^ { - n x } \right)$ 
E)  $y = c \left( e ^ { n x } + e ^ { - n x } \right)$ 
A)  $y = 0$ 
B)  $y = c \cdot \sin ( n x )$ 
C)  $y = c \cdot \cos ( n x )$ 
D)  $y = c \left( e ^ { n x } - e ^ { - n x } \right)$ 
E)  $y = c \left( e ^ { n x } + e ^ { - n x } \right)$

Question 7

After separating variables in the previous problem, the eigenvalue problem becomes

Accepted Answer

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

Question 8

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { xx } + u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

Accepted Answer

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

Question 9

The solution of $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ if $\lambda = 0$ is

Accepted Answer

A)  $y = \sin x$ 
B)  $y = \cos x$ 
C)  $y = 0$ 
D)  $y = a x + b$ 
E)  $y = x - \pi$ 
A)  $y = \sin x$ 
B)  $y = \cos x$ 
C)  $y = 0$ 
D)  $y = a x + b$ 
E)  $y = x - \pi$

Question 10

The solution of the eigenvalue problem from the previous problem is

Accepted Answer

A)  $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
B)  $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$ 
E)  $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$ 
A)  $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
B)  $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$ 
E)  $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$

Question 11

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = \sin u$ is Select all that apply.

Accepted Answer

A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic 
A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic

Question 12

In the previous two problems, the product solutions are

Accepted Answer

A)  $\sin ( n \pi x / L ) \sin ( n \pi / L )$ 
B)  $\sin ( n \pi x / L ) \cos ( n \pi / L )$ 
C)  $\cos ( n \pi x / L ) \cos ( n \pi / L )$ 
D)  $\sin ( n \pi x / L ) e ^ { n x / L }$ 
E)  $\cos ( n \pi x / L ) e ^ { - n t / L }$ 
A)  $\sin ( n \pi x / L ) \sin ( n \pi / L )$ 
B)  $\sin ( n \pi x / L ) \cos ( n \pi / L )$ 
C)  $\cos ( n \pi x / L ) \cos ( n \pi / L )$ 
D)  $\sin ( n \pi x / L ) e ^ { n x / L }$ 
E)  $\cos ( n \pi x / L ) e ^ { - n t / L }$

Question 13

In the previous three problems, the solution of the original problem is

Accepted Answer

A)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
B)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \cos ( n \pi t /L )$ , where $c _ { 0 } = \int _ { 0 } ^ { L } f ( x ) d x / L , c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
C)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sin ( n \pi t / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
D)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - nxt / L }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - n x t / L }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
A)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$ 
B)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \cos ( n \pi t /L )$ , where $c _ { 0 } = \int _ { 0 } ^ { L } f ( x ) d x / L , c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
C)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sin ( n \pi t / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$ 
D)  $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - nxt / L }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - n x t / L }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$

Question 14

In the previous problem, the eigenfunction expansion if $x e ^ { t }$ is

Accepted Answer

A)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / ( n \pi )$ 
B)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \sin ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
C)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
D)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \cos ( n \pi x / L ) / ( n \pi )$ 
E)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \sin ( n \pi x / L ) / ( n \pi )$ 
A)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / ( n \pi )$ 
B)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \sin ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
C)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
D)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \cos ( n \pi x / L ) / ( n \pi )$ 
E)  $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \sin ( n \pi x / L ) / ( n \pi )$

Question 15

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = u$ is Select all that apply.

Accepted Answer

A)  nonlinear 
B)  linear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic 
A)  nonlinear 
B)  linear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic

Question 16

In the previous three problems, the solution of the original problem is

Accepted Answer

A)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
B)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
C)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x$ 
D)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n - 0 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x$ 
A)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
B)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
C)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x$ 
D)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$ 
E)  $u ( x , t ) = \sum _ { n - 0 } ^ { \infty } c _ { n } \sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x$

Question 17

In the previous three problems, the solution of the original problem is

Accepted Answer

A)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$ 
B)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
C)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh ( n \pi ( y - H ) / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
D)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \sinh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$ 
A)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$ 
B)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
C)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh ( n \pi ( y - H ) / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
D)  $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$ 
E)  $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \sinh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$

Question 18

Consider the equation $u _ { xx } - u _ { t } = 0$ with conditions $u ( 0 , t ) = 0 , u _ { x } = ( L , t ) = 0 , u ( x , 0 ) = f ( x )$ . When separating variables with $u ( x , t ) = X ( x ) T ( t )$ , the resulting problems for $X , T$ are

Accepted Answer

A)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } - \lambda T = 0 , T ( 0 ) = f ( x )$ 
B)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0 , T ( 0 ) = f ( x )$ 
C)  $X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0$ 
D)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } - \lambda T = 0$ 
E)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0$ 
A)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } - \lambda T = 0 , T ( 0 ) = f ( x )$ 
B)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0 , T ( 0 ) = f ( x )$ 
C)  $X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0$ 
D)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } - \lambda T = 0$ 
E)  $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime } + \lambda T = 0$

Question 19

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

Accepted Answer

A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic 
A)  linear 
B)  nonlinear 
C)  hyperbolic 
D)  elliptic 
E)  parabolic

Question 20

In the previous three problems, the solution for $u ( t )$ is

Accepted Answer

A)  $u _ { n } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } a ^ { 2 } t / L ^ { 2 } } \right) / ( n \pi ( 1 + n \pi / L ) )$ 
B)  $u _ { x } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
C)  $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
D)  $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
E)  $u _ { n } ( t ) = 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
A)  $u _ { n } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } a ^ { 2 } t / L ^ { 2 } } \right) / ( n \pi ( 1 + n \pi / L ) )$ 
B)  $u _ { x } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
C)  $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
D)  $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$ 
E)  $u _ { n } ( t ) = 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$

In the previous three problems, the solution of the original problem is

In the previous two problems, the product solutions are

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { x } - u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u$ is

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

The solution of $y ^ { \prime \prime } - n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldots$ is

After separating variables in the previous problem, the eigenvalue problem becomes

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { xx } + u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

The solution of $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ if $\lambda = 0$ is

The solution of the eigenvalue problem from the previous problem is

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = \sin u$ is Select all that apply.

In the previous two problems, the product solutions are

In the previous three problems, the solution of the original problem is

In the previous problem, the eigenfunction expansion if $x e ^ { t }$ is

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = u$ is Select all that apply.

In the previous three problems, the solution of the original problem is

In the previous three problems, the solution of the original problem is

Consider the equation $u _ { xx } - u _ { t } = 0$ with conditions $u ( 0 , t ) = 0 , u _ { x } = ( L , t ) = 0 , u ( x , 0 ) = f ( x )$ . When separating variables with $u ( x , t ) = X ( x ) T ( t )$ , the resulting problems for $X , T$ are

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

In the previous three problems, the solution for $u ( t )$ 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 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 12: Boundary-Value Problems in Rectangular Coordinates

In the previous three problems, the solution of the original problem is

In the previous two problems, the product solutions are

When u(x,y)=X(x)Y(y)u ( x , y ) = X ( x ) Y ( y )u(x,y)=X(x)Y(y) is substituted into the equation ux−uyy=0u _ { x } - u _ { y y } = 0ux​−uyy​=0 , the resulting equations for XXX and YYY are

The differential equation ∂2u∂x2+∂2u∂y2=u\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u∂x2∂2u​+∂y2∂2u​=u is

The differential equation ∂2u∂x2+∂2u∂y2=u\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u∂x2∂2u​+∂y2∂2u​=u is Select all that apply.

The solution of y′′−n2y=0,y(0)=0,y(π)=0,n=1,2,3…y ^ { \prime \prime } - n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldotsy′′−n2y=0,y(0)=0,y(π)=0,n=1,2,3… is

After separating variables in the previous problem, the eigenvalue problem becomes

When u(x,y)=X(x)Y(y)u ( x , y ) = X ( x ) Y ( y )u(x,y)=X(x)Y(y) is substituted into the equation uxx+uyy=0u _ { xx } + u _ { y y } = 0uxx​+uyy​=0 , the resulting equations for XXX and YYY are

The solution of y′′+λy=0,y(0)=0,y(π)=0y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0y′′+λy=0,y(0)=0,y(π)=0 if λ=0\lambda = 0λ=0 is

The solution of the eigenvalue problem from the previous problem is

The differential equation ∂2u∂x2−∂2u∂y2=sin⁡u\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = \sin u∂x2∂2u​−∂y2∂2u​=sinu is Select all that apply.

In the previous two problems, the product solutions are

In the previous three problems, the solution of the original problem is

In the previous problem, the eigenfunction expansion if xetx e ^ { t }xet is

The differential equation ∂2u∂x2−∂u∂t=u\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = u∂x2∂2u​−∂t∂u​=u is Select all that apply.

In the previous three problems, the solution of the original problem is

In the previous three problems, the solution of the original problem is

In the previous three problems, the solution for u(t)u ( t )u(t) 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 Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { x } - u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = u$ is

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

The solution of $y ^ { \prime \prime } - n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldots$ is

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { xx } + u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

The solution of $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ if $\lambda = 0$ is

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = \sin u$ is Select all that apply.

In the previous problem, the eigenfunction expansion if $x e ^ { t }$ is

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = u$ is Select all that apply.

In the previous three problems, the solution for $u ( t )$ is