Exam 12: Boundary-Value Problems in Rectangular Coordinates

The wave equation for a vibrating string is derived using the assumptions Select all that apply.

The quantity of heat in an element of a rod of mass $m$ is proportional to Select all that apply.

The solution of the eigenvalue problem from the previous problem is

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

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

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

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

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

The solution of the eigenvalue problem from the previous problem is

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

In the previous two problems, the solution for $u$ takes the form

The differential equation $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$ is

The solution of the eigenvalue problem in the previous problem is

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

The solution of the eigenvalue problem from the previous problem is

In the previous two problems, the product solutions are

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

In the previous two problems, the product solutions are

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

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