Question 1

The general solution of $y ^ { \prime \prime } + n ^ { 2 } n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0 , n = 1,2,3 \ldots$ is&#10;A) $y = 0$&#10;B) $y = c \cdot \sin ( n \pi x )$&#10;C) $y = c \cdot \cos ( n \pi x )$&#10;D) $c \left( e ^ { n + x } - e ^ { - n + x } \right)$&#10;E) $c \left( e ^ { n + x } + e ^ { - n + x } \right)$

Accepted Answer

The given differential equation is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation of such a differential equation is $r^2 + n^4 = 0$, which has solutions $r = \pm in^2$. The general solution to the differential equation is therefore a combination of sine and cosine functions. Given the boundary conditions $y(0) = 0$ and $y(1) = 0$, the cosine term is eliminated because $y = c \cdot \cos(n \pi x)$ does not satisfy $y(0) = 0$ for any non-zero $c$. The sine function, $y = c \cdot \sin(n \pi x)$, satisfies both boundary conditions for $n = 1,2,3,\ldots$ because $\sin(n \pi) = 0$ for any integer $n$, including at $x = 0$ and $x = 1$.

Question 2

The solution of the eigenvalue problem from the previous problem is&#10;A) $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;B) $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;C) $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;D) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;E) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$

Accepted Answer

$\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;

Question 3

The wave equation for a vibrating string is derived using the assumptions Select all that apply.&#10;A) the string is perfectly flexible.&#10;B) the displacements may be large.&#10;C) the tension acts perpendicular to the string.&#10;D) the tension is large compared with gravity.&#10;E) the string is homogeneous.

Accepted Answer

The wave equation for a vibrating string assumes that the string is perfectly flexible (A), meaning it can bend without resistance but cannot stretch, and that it is homogeneous (E), implying uniform density and tension throughout. The tension in the string must be large compared with the force of gravity (D) to ensure that the gravitational force does not significantly affect the vibration. Assumptions B and C are incorrect because the displacements are assumed to be small (not large) for linear approximation to hold, and the tension acts tangentially (along the string) rather than perpendicularly.

Question 4

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.

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

Accepted Answer

The given differential equation is a second order (since the highest derivative is of second order), linear (since the equation involves only linear terms of the unknown function and its derivatives), homogeneous (since there is no term that is independent of the function $u$).

Question 5

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.

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

Accepted Answer

The given differential equation is linear because it can be expressed in the form $Lu = f$, where $L$ is a linear operator. It is parabolic because it resembles the heat equation form, which is a standard parabolic equation.

Question 6

In the previous three problems, the solution of the original problem is&#10;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 $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x / L$&#10;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 $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$&#10;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 $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( ( n - 1 / 2 ) \pi x / L ) d x$&#10;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 $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x / L$&#10;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 $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( ( n - 1 / 2 ) \pi x / L ) d x$

Accepted Answer

The answer of In the previous three problems, the solution...

Question 7

The solution of the eigenvalue problem from the previous problem is&#10;A) $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;B) $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;C) $\lambda = ( n - 1 / 2 ) \pi / L , X ( x ) = \sin ( ( n - 1 / 2 ) \pi x / L ) , n = 1,2,3 , \ldots$&#10;D) $\lambda = ( ( n - 1 / 2 ) \pi / L ) ^ { 2 } , X ( x ) = \sin ( ( n - 1 / 2 ) \pi x / L ) , n = 1,2,3 , \ldots$&#10;E) $\lambda = ( ( n - 1 / 2 ) \pi / L ) ^ { 2 } , X ( x ) = \cos ( ( n - 1 / 2 ) \pi x / L ) , n = 1,2,3 , \ldots$

Accepted Answer

The correct solution involves squaring the term $( ( n - 1 / 2 ) \pi / L )$ to get the eigenvalue $\lambda$, and using the sine function for $X(x)$ with $n = 1,2,3, \ldots$, which corresponds to the boundary conditions typically found in problems involving a string fixed at both ends or similar physical systems.

Question 8

In the previous three problems, the solution of the original problem is&#10;A) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$&#10;B) $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) \cos ( n \pi t /L )$ , where $c _ { 0 } = \int _ { 0 } ^ { L } f ( x ) d x / L , c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$&#10;C) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sin ( n \pi t / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$&#10;D) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - nxt / L }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$&#10;E) $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - n x t / L }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$

Accepted Answer

The answer of In the previous three problems, the solution...

Question 9

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.

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

Accepted Answer

The given differential equation is linear because it involves linear operations on the function $u$ and its derivatives (no terms involve powers or products of $u$ or its derivatives). It is elliptic because the discriminant of the characteristic equation derived from the highest order partial derivatives, $B^2 - 4AC$, where $A = 1$, $B = 1$, and $C = 1$, is negative ($1^2 - 4(1)(1) = -3$).

Question 10

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

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

Accepted Answer

Substituting $u(x,t) = X(x)T(t)$ into the given PDE $u_{xx} - u_t = 0$ and dividing by $XT$ leads to $\frac{X''}{X} = \frac{T'}{T}$, implying both sides equal a constant $\lambda$. The boundary conditions for $X(x)$ are directly given as $X(0) = 0$ and $X'(L) = 0$. The condition $T'(0) = f(x)$ is not applicable since $T$ is a function of $t$ only, not $x$. Thus, the correct separation leads to $X'' + \lambda X = 0$ with the given boundary conditions for $X$, and $T' + \lambda T = 0$ for $T$, without an initial condition involving $f(x)$ directly in the separated equations.

Question 11

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

A)

X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ( 0 ) = 0

B)

X ^ { \prime \prime } - \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0

C)

X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0

D)

X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime \prime } - \lambda T = 0 , T ^ { \prime } ( 0 ) = 0

E)

X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( L ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ( 0 ) = f ( x )

Accepted Answer

The answer of Consider the equation \(u _ { xx...

Question 12

In the previous two problems, the product solutions are&#10;A) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$&#10;B) $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$&#10;C) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) / L }$&#10;D) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$&#10;E) $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$

Accepted Answer

The answer of In the previous two problems, the product...

Question 13

The solution of the previous three problems is $\sum _ { n = 1 } ^ { \infty } c _ { n } X _ { n } ( x ) e ^ { - \lambda _ { n } k t }$ , where $X _ { n }$ and $\lambda _ { n }$ are given in the previous problem and

A)

c _ { n } = \int _ { 0 } ^ { L } f ( x ) d x

B)

c _ { n } = \int _ { 0 } ^ { L } f ( x ) X _ { n } ( x ) d x

C)

c _ { n } = \int _ { 0 } ^ { L } f ( x ) X _ { n } ( x ) d x / \int _ { 0 } ^ { L } X _ { n } ( x ) d x

D)

c _ { n } = \int _ { 0 } ^ { L } X _ { n } ^ { 2 } ( x ) d x / \int _ { 0 } ^ { L } f ( x ) X _ { n } ( x ) d x

E)

c _ { n } = \int _ { 0 } ^ { L } f ( x ) X _ { n } ( x ) d x / \int _ { 0 } ^ { L } X _ { n } ^ { 2 } ( x ) d x

Accepted Answer

The answer of The solution of the previous three problems...

Question 14

After separating variables in the previous problem, the eigenvalue problem becomes&#10;A) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0$&#10;B) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = - h X ( L )$&#10;C) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = - h X ^ { \prime } ( 0 )$&#10;D) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = 0$&#10;E) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ^ { \prime } ( L ) = h X ( L )$

Accepted Answer

The answer of After separating variables in the previous problem,...

Question 15

In the previous two problems, the product solutions are&#10;A) $\sin ( n \pi x / L ) \sin ( n \pi / L )$&#10;B) $\sin ( n \pi x / L ) \cos ( n \pi / L )$&#10;C) $\cos ( n \pi x / L ) \cos ( n \pi / L )$&#10;D) $\sin ( n \pi x / L ) e ^ { n x / L }$&#10;E) $\cos ( n \pi x / L ) e ^ { - n t / L }$

Accepted Answer

The answer of In the previous two problems, the product...

Question 16

The solution of $y ^ { \prime \prime } - n ^ { 2 } n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0 , n = 1,2,3 \ldots$ is&#10;A) $y = 0$&#10;B) $y = c \cdot \sin ( n \pi x )$&#10;C) $y = c \cdot \cos ( n \pi x )$&#10;D) $c \left( e ^ { n xt } - e ^ { - nxt } \right)$&#10;E) $c \left( e ^ { n xt } + e ^ { - nxt } \right)$

Accepted Answer

The answer of The solution of \(y ^ { \prime...

Question 17

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&#10;A) $X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } + \lambda Y = 0$&#10;B) $X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } - \lambda Y = 0$&#10;C) $X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0$&#10;D) $X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } - \lambda Y = 0$&#10;E) $X ^ { \prime \prime } - \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0$

Accepted Answer

The answer of When \(u ( x , y )...

Question 18

The solution of the eigenvalue problem in the previous problem is&#10;A) $\lambda = ( ( n - 1 / 2 ) \pi / L ) ^ { 2 } , X = \sin ( ( n - 1 / 2 ) \pi x / L )$&#10;B) $\sqrt { \lambda } / h = \tan ( \sqrt { \lambda } L ) , X = \sin ( \sqrt { \lambda } x )$&#10;C) $\lambda = ( n \pi / L ) ^ { 2 } , X = \sin ( n \pi x / L )$&#10;D) $\sqrt { \lambda } / h = - \tan ( \sqrt { \lambda } L ) , X = \sin ( \sqrt { \lambda } x )$&#10;E) $\sqrt { \lambda } h = - \tan ( \sqrt { \lambda } L ) , X = \sin ( \sqrt { \lambda } x )$

Accepted Answer

The answer of The solution of the eigenvalue problem in...

Question 19

The solution of $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ if $\lambda = 0$ is&#10;A) $y = \sin x$&#10;B) $y = \cos x$&#10;C) $y = 0$&#10;D) $y = a x + b$&#10;E) $y = x - \pi$

Accepted Answer

The answer of The solution of \(y ^ { \prime...

Question 20

The model describing the temperature in a rod where the temperature at the left end is zero and where there is heat transfer from the right boundary into the external medium is&#10;A) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$&#10;B) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$&#10;C) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$&#10;D) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$&#10;E) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 0 , t ) = 0 , \frac { \partial u } { \partial x } = ( L , t ) = - h u ( L , t ) , h > 0 , u ( x , 0 ) = f ( x )$

Accepted Answer

The answer of The model describing the temperature in a...

Question 21

Consider the equation $u _ { x } + u _ { y y } = 0$ with conditions $u ( 0 , y ) = 0 , u = ( L , y ) = 0 , u ( x , 0 ) = f ( x ) , u ( x , H ) = 0$ . When separating variables with $u ( x , y ) = X ( x ) Y ( y )$ , the resulting problems for $X , Y$ are

A)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( H ) = 0

B)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = 0

C)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0

D)

X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = f ( 0 )

E)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = f ( x )

Accepted Answer

The answer of Consider the equation \(u _ { x...

Question 22

In the previous problem, the eigenfunction expansion if $x e ^ { t }$ is&#10;A) $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / ( n \pi )$&#10;B) $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \sin ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$&#10;C) $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 L \cos ( n \pi x / L ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$&#10;D) $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \cos ( n \pi x / L ) / ( n \pi )$&#10;E) $e ^ { t } \sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 L \sin ( n \pi x / L ) / ( n \pi )$

Accepted Answer

The answer of In the previous problem, the eigenfunction expansion...

Question 23

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&#10;A) $X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0$&#10;B) $X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } - \lambda Y = 0$&#10;C) $X ^ { \prime \prime } - \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0$&#10;D) $X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } + \lambda Y = 0$&#10;E) $X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } - \lambda Y = 0$

Accepted Answer

The answer of When \(u ( x , y )...

Question 24

The solution of the eigenvalue problem from the previous problem is&#10;A) $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;B) $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;C) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;D) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;E) $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$

Accepted Answer

The answer of The solution of the eigenvalue problem from...

Question 25

The solution of the eigenvalue problem from the previous problem is&#10;A) $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;B) $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;C) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$&#10;D) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$&#10;E) $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$

Accepted Answer

The answer of The solution of the eigenvalue problem from...

Question 26

The solution of $y ^ { \prime \prime } - n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldots$ is&#10;A) $y = 0$&#10;B) $y = c \cdot \sin ( n x )$&#10;C) $y = c \cdot \cos ( n x )$&#10;D) $y = c \left( e ^ { n x } - e ^ { - n x } \right)$&#10;E) $y = c \left( e ^ { n x } + e ^ { - n x } \right)$

Accepted Answer

The answer of The solution of \(y ^ { \prime...

Question 27

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.

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

Accepted Answer

The answer of The differential equation \(\frac { \partial ^...

Question 28

In the previous three problems, the solution of the original problem is&#10;A) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$&#10;B) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$&#10;C) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) \cosh ( n \pi ( y - H ) / L )$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$&#10;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 $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / ( L \sinh ( - n \pi H / L ) )$&#10;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 $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / \sinh ( - n \pi H / L )$

Accepted Answer

The answer of In the previous three problems, the solution...

Question 29

In the previous two problems, the product solutions are&#10;A) $\sin ( n \pi x / L ) \sinh ( n \pi ( y - H ) / L )$&#10;B) $\cos ( n \pi x / L ) \sinh ( - n \pi ( y - H ) / L )$&#10;C) $\cos ( n \pi x / L ) \cosh \left( ( n \pi / L ) ^ { 2 } ( y - H ) \right)$&#10;D) $\sin ( n \pi x / L ) \cosh \left( - ( n \pi / L ) ^ { 2 } ( y - H ) \right)$&#10;E) $\cos ( n \pi x / L ) \cosh ( - n \pi ( y - H ) / L )$

Accepted Answer

The answer of In the previous two problems, the product...

Question 30

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.

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

Accepted Answer

The answer of The differential equation \(\frac { \partial ^...

Question 31

In the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + x e ^ { t } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( L , t ) = 0 , u ( x , 0 ) = 0$ , the eigenvalues and eigenfunctions of the underlying homogeneous problem are

A)

\lambda = n ^ { 2 } , X = \sin ( n x )

B)

\lambda = n ^ { 2 } , X = \cos ( n x )

C)

\lambda = n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } , X = \sin ( n \pi x / L )

D)

\lambda = n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } , X = \cos ( n \pi x / L )

E)

\lambda = n ^ { 2 } \pi ^ { 2 } , X = \sin ( n \pi x )

Accepted Answer

The answer of In the problem \(\frac { \partial ^...

Question 32

The quantity of heat in an element of a rod of mass $m$ is proportional to Select all that apply.&#10;A) mass&#10;B) thermal conductivity&#10;C) specific heat&#10;D) thermal diffusivity&#10;E) temperature

Accepted Answer

The answer of The quantity of heat in an element...

Question 33

In the previous two problems, the solution for $u$ takes the form&#10;A) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } u _ { n } ( t ) \sin ( n \pi x / L )$&#10;B) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } u _ { n } ( t ) \cos ( n \pi x )$&#10;C) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } u _ { n } ( t ) \sin \left( n ^ { 2 } \pi ^ { 2 } x ^ { 2 } / L ^ { 2 } \right)$&#10;D) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } u _ { n } ( t ) \cos \left( n ^ { 2 } \pi ^ { 2 } x ^ { 2 } / L ^ { 2 } \right)$&#10;E) $u ( x , t ) = \sum _ { n - 1 } ^ { \infty } u _ { n } ( t ) \sin ( n \pi x )$

Accepted Answer

The answer of In the previous two problems, the solution...

Question 34

In the previous two problems, the product solutions are&#10;A) $\sin ( n \pi x / L ) e ^ { n x / L }$&#10;B) $\sin ( n \pi x / L ) e ^ { - n x / L }$&#10;C) $\cos ( n \pi x / L ) e ^ { ( n \pi / D ) ^ { 2 } t }$&#10;D) $\sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$&#10;E) $\cos ( n \pi x / L ) e ^ { - n t / L }$

Accepted Answer

The answer of In the previous two problems, the product...

Question 35

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

A)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { t } + \lambda T = 0 , T ( 0 ) = f ( 0 )

B)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { t } + \lambda T = 0 , T ( 0 ) = f ( x )

C)

X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { \prime } + \lambda T = 0

D)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { \prime } - \lambda T = 0

E)

X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { \prime } + \lambda T = 0

Accepted Answer

The answer of Consider the equation \(u _ { m...

Question 36

In the previous three problems, the solution of the original problem is&#10;A) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$&#10;B) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x$&#10;C) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x / L$&#10;D) $u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { n } = 2 \int _ { 0 } ^ { L } f ( x ) \sin ( n \pi x / L ) d x / L$&#10;E) $u ( x , t ) = \sum _ { n = 0 } ^ { \infty } c _ { n } \cos ( n \pi x / L ) e ^ { - ( n \pi / L ) ^ { 2 } t }$ , where $c _ { x } = 2 \int _ { 0 } ^ { L } f ( x ) \cos ( n \pi x / L ) d x$

Accepted Answer

The answer of In the previous three problems, the solution...

Question 37

The general solution of $y ^ { \prime \prime } + n ^ { 2 } y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0 , n = 1,2,3 \ldots$ is&#10;A) $y = 0$&#10;B) $y = c \cdot \sin ( n x )$&#10;C) $y = c \cdot \cos ( n x )$&#10;D) $y = c \left( e ^ { n x } - e ^ { - n x } \right)$&#10;E) $y = c \left( e ^ { n x } + e ^ { - n x } \right)$

Accepted Answer

The answer of The general solution of \(y ^ {...

Question 38

The solution of $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ if $\lambda = 0$ is&#10;A) $y = a x + b$&#10;B) $y = x - \pi$&#10;C) $y = 0$&#10;D) $y = \sin x$&#10;E) $y = \cos x$

Accepted Answer

The answer of The solution of \(y ^ { \prime...

Question 39

In the previous three problems, the solution for $u ( t )$ is&#10;A) $u _ { n } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } a ^ { 2 } t / L ^ { 2 } } \right) / ( n \pi ( 1 + n \pi / L ) )$&#10;B) $u _ { x } ( t ) = ( - 1 ) ^ { n } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$&#10;C) $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } + e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$&#10;D) $u _ { x } ( t ) = ( - 1 ) ^ { n + 1 } 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$&#10;E) $u _ { n } ( t ) = 2 L \left( e ^ { t } - e ^ { - n ^ { 2 } x ^ { 2 } t / L ^ { 2 } } \right) / \left( n \pi \left( 1 + n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } \right) \right)$

Accepted Answer

The answer of In the previous three problems, the solution...

Question 40

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.

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

Accepted Answer

The answer of The differential equation \(\frac { \partial ^...

Deck 12: Boundary-Value Problems in Rectangular Coordinates