Question 1

In the previous problem, the solution of the eigenvalue problem is

Accepted Answer

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

Question 2

In the previous two problems, the product solutions are

Accepted Answer

A)  $u _ { n } = r ^ { n } P _ { n } ( \cos \theta )$ 
B)  $u _ { n } = r ^ { - n } P _ { n } ( \cos \theta )$ 
C)  $u _ { n } = r ^ { n } P _ { n } ( \sin \theta )$ 
D)  $u _ { n } = r ^ { - n } P _ { n } ( \sin \theta )$ 
E)  none of the above 
A)  $u _ { n } = r ^ { n } P _ { n } ( \cos \theta )$ 
B)  $u _ { n } = r ^ { - n } P _ { n } ( \cos \theta )$ 
C)  $u _ { n } = r ^ { n } P _ { n } ( \sin \theta )$ 
D)  $u _ { n } = r ^ { - n } P _ { n } ( \sin \theta )$ 
E)  none of the above

Question 3

The Laplacian in polar coordinates is

Accepted Answer

A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$

Question 4

The solution of $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , R ( 1 ) = 0$ is

Accepted Answer

A)  $\lambda = n \pi , R = \sin ( n \pi \ln r )$ 
B)  $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r )$ 
C)  $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r ) + \cos ( n \pi \ln r )$ 
D)  $\lambda = n \pi , R = \sin ( n \pi \ln r ) - \cos ( n \pi \ln r )$ 
E)  none of the above 
A)  $\lambda = n \pi , R = \sin ( n \pi \ln r )$ 
B)  $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r )$ 
C)  $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r ) + \cos ( n \pi \ln r )$ 
D)  $\lambda = n \pi , R = \sin ( n \pi \ln r ) - \cos ( n \pi \ln r )$ 
E)  none of the above

Question 5

In the previous three problems, the product solutions are

Accepted Answer

A)  $u = J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$ 
B)  $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$ 
C)  $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$ 
D)  $u = J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$ 
E)  $u = J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$ 
A)  $u = J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$ 
B)  $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$ 
C)  $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$ 
D)  $u = J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$ 
E)  $u = J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$

Question 6

In the previous problem, after separating variables, the resulting problems are

Accepted Answer

A)  $r R ^ { \prime \prime } - R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$ 
B)  $r R ^ { \prime \prime } + R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
C)  $r R ^ { \prime \prime } - R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
D)  $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
E)  $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$ 
A)  $r R ^ { \prime \prime } - R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$ 
B)  $r R ^ { \prime \prime } + R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
C)  $r R ^ { \prime \prime } - R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
D)  $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$ 
E)  $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$

Question 7

Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary. The mathematical model of this situation is

Accepted Answer

A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$

Question 8

In the previous four problems, the infinite series solution is (for certain values of the constants)

Accepted Answer

A)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$ 
B)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$ 
C)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$ 
D)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$ 
E)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$ 
A)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$ 
B)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$ 
C)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$ 
D)  $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$ 
E)  $u = \sum _ { n = 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$

Question 9

In changing from Cartesian to polar coordinates, $\frac { \partial u } { \partial x }$ is

Accepted Answer

A)  $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$ 
B)  $\sin \theta \frac { \partial u } { \partial r } - \cos \theta \frac { \partial u } { \partial \theta } / r$ 
C)  $\cos \theta \frac { \partial u } { \partial r } - \sin \theta \frac { \partial u } { \partial \theta } / r$ 
D)  $\cos \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$ 
E)  $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta }$ 
A)  $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$ 
B)  $\sin \theta \frac { \partial u } { \partial r } - \cos \theta \frac { \partial u } { \partial \theta } / r$ 
C)  $\cos \theta \frac { \partial u } { \partial r } - \sin \theta \frac { \partial u } { \partial \theta } / r$ 
D)  $\cos \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$ 
E)  $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta }$

Question 10

In the previous problem, after separating variables using $u ( r , t ) = R ( r ) T ( t )$ . the resulting problems are

Accepted Answer

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

Question 11

In the three previous problems, the product solutions are

Accepted Answer

A)  $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 0,1,2 , \ldots$ 
B)  $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 1,2,3 , \ldots$ 
C)  $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$ 
D)  $u _ { n } = r ^ { n } \cos ( n \theta ) , n = 0,1,2 , \ldots$ 
E)  $u _ { n } = r ^ { n } \sin ( n \theta ) , n = 1,2,3 , \ldots$ 
A)  $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 0,1,2 , \ldots$ 
B)  $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 1,2,3 , \ldots$ 
C)  $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$ 
D)  $u _ { n } = r ^ { n } \cos ( n \theta ) , n = 0,1,2 , \ldots$ 
E)  $u _ { n } = r ^ { n } \sin ( n \theta ) , n = 1,2,3 , \ldots$

Question 12

The two dimensional wave equation in polar coordinates is

Accepted Answer

A)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
B)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
C)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
D)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
E)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } - \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
A)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
B)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
C)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
D)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ 
E)  $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } - \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$

Question 13

Consider the steady-state temperature distribution in a circular cylinder of radius 2 and height 3, with zero temperature at $r = 2$ and at $z = 3$ and a temperature of 10 at $z = 0$ . The mathematical model for this problem is

Accepted Answer

A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$

Question 14

In the previous problem, the solution of the eigenvalue problem is

Accepted Answer

A)  $\lambda = z _ { n } ^ { 2 } / 4$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r / 2 \right)$ 
B)  $\lambda = z _ { n } ^ { 2 }$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r \right)$ 
C)  $\lambda = n \pi / 9 , Z = \cos ( n \pi z / 3 )$ 
D)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \cos ( n \pi z / 3 )$ 
E)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \sin ( n \pi z / 3 )$ 
A)  $\lambda = z _ { n } ^ { 2 } / 4$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r / 2 \right)$ 
B)  $\lambda = z _ { n } ^ { 2 }$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r \right)$ 
C)  $\lambda = n \pi / 9 , Z = \cos ( n \pi z / 3 )$ 
D)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \cos ( n \pi z / 3 )$ 
E)  $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \sin ( n \pi z / 3 )$

Question 15

In the previous problem, separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems are

Accepted Answer

A)  $R ^ { \prime \prime } + r R ^ { \prime } - r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
B)  $R ^ { \prime \prime } + r R ^ { \prime } + r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
C)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } - \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
D)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
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)  $R ^ { \prime \prime } + r R ^ { \prime } - r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
B)  $R ^ { \prime \prime } + r R ^ { \prime } + r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
C)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } - \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
D)  $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$ 
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 16

In the previous four problems, the infinite series solution of the original problem is $u = \sum _ { n = 1 } ^ { \infty } r ^ { n } \left( A _ { n } \cos ( n \theta ) + B _ { n } \sin ( n \theta ) \right)$ where Select all that apply.

Accepted Answer

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

Question 17

The relationships between Cartesian and spherical coordinates are Select all that apply.

Accepted Answer

A)  $r ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ 
B)  $\tan \phi = y / x$ 
C)  $x = r \cos \phi \sin \theta$ 
D)  $y = r \cos \phi \cos \theta$ 
E)  $z = r \cos \theta$ 
A)  $r ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ 
B)  $\tan \phi = y / x$ 
C)  $x = r \cos \phi \sin \theta$ 
D)  $y = r \cos \phi \cos \theta$ 
E)  $z = r \cos \theta$

Question 18

In changing from Cartesian to spherical coordinates, $\frac { \partial r } { \partial x }$ becomes

Accepted Answer

A)  $\sin \phi \sin \theta$ 
B)  $\sin \phi \cos \theta$ 
C)  $\cos \phi \sin \theta$ 
D)  $\cos \phi \cos \theta$ 
E)  $\sin \phi$ 
A)  $\sin \phi \sin \theta$ 
B)  $\sin \phi \cos \theta$ 
C)  $\cos \phi \sin \theta$ 
D)  $\cos \phi \cos \theta$ 
E)  $\sin \phi$

Question 19

In the previous problem, the solution of the eigenvalue problem is

Accepted Answer

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

Question 20

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of $1 - \sin ( \pi r / 2 )$ and an initial velocity of zero. The mathematical model for this situation is

Accepted Answer

A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi / 2 ) , u _ { t } ( r , 0 ) = 0$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u _ { t } ( r , 0 ) = 0$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$ 
A)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$ 
B)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$ 
C)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi / 2 ) , u _ { t } ( r , 0 ) = 0$ 
D)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u _ { t } ( r , 0 ) = 0$ 
E)  $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$

In the previous problem, the solution of the eigenvalue problem is

In the previous two problems, the product solutions are

The Laplacian in polar coordinates is

The solution of $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , R ( 1 ) = 0$ is

In the previous three problems, the product solutions are

In the previous problem, after separating variables, the resulting problems are

Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary. The mathematical model of this situation is

In the previous four problems, the infinite series solution is (for certain values of the constants)

In changing from Cartesian to polar coordinates, $\frac { \partial u } { \partial x }$ is

In the previous problem, after separating variables using $u ( r , t ) = R ( r ) T ( t )$ . the resulting problems are

In the three previous problems, the product solutions are

The two dimensional wave equation in polar coordinates is

Consider the steady-state temperature distribution in a circular cylinder of radius 2 and height 3, with zero temperature at $r = 2$ and at $z = 3$ and a temperature of 10 at $z = 0$ . The mathematical model for this problem is

In the previous problem, the solution of the eigenvalue problem is

In the previous problem, separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems are

In the previous four problems, the infinite series solution of the original problem is $u = \sum _ { n = 1 } ^ { \infty } r ^ { n } \left( A _ { n } \cos ( n \theta ) + B _ { n } \sin ( n \theta ) \right)$ where Select all that apply.

The relationships between Cartesian and spherical coordinates are Select all that apply.

In changing from Cartesian to spherical coordinates, $\frac { \partial r } { \partial x }$ becomes

In the previous problem, the solution of the eigenvalue problem is

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of $1 - \sin ( \pi r / 2 )$ and an initial velocity of zero. The mathematical model for this situation 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 Rectangular Coordinates

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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Exam 13: Boundary-Value Problems in Other Coordinate Systems

In the previous problem, the solution of the eigenvalue problem is

In the previous two problems, the product solutions are

The Laplacian in polar coordinates is

The solution of r2R′′+rR′+λR=0,R(0)=0,R(1)=0r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , R ( 1 ) = 0r2R′′+rR′+λR=0,R(0)=0,R(1)=0 is

In the previous three problems, the product solutions are

In the previous problem, after separating variables, the resulting problems are

Consider the steady-state temperature distribution in a circular disc of radius CCC centere at the origin, with temperature given as a function, f(θ)f ( \theta )f(θ) on the boundary. The mathematical model of this situation is

In the previous four problems, the infinite series solution is (for certain values of the constants)

In changing from Cartesian to polar coordinates, ∂u∂x\frac { \partial u } { \partial x }∂x∂u​ is

In the previous problem, after separating variables using u(r,t)=R(r)T(t)u ( r , t ) = R ( r ) T ( t )u(r,t)=R(r)T(t) . the resulting problems are

In the three previous problems, the product solutions are

The two dimensional wave equation in polar coordinates is

Consider the steady-state temperature distribution in a circular cylinder of radius 2 and height 3, with zero temperature at r=2r = 2r=2 and at z=3z = 3z=3 and a temperature of 10 at z=0z = 0z=0 . The mathematical model for this problem is

In the previous problem, the solution of the eigenvalue problem is

In the previous problem, separate variables using u(r,θ)=R(r)Θ(θ)u ( r , \theta ) = R ( r ) \Theta ( \theta )u(r,θ)=R(r)Θ(θ) . The resulting problems are

The relationships between Cartesian and spherical coordinates are Select all that apply.

In changing from Cartesian to spherical coordinates, ∂r∂x\frac { \partial r } { \partial x }∂x∂r​ becomes

In the previous problem, the solution of the eigenvalue problem is

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of 1−sin⁡(πr/2)1 - \sin ( \pi r / 2 )1−sin(πr/2) and an initial velocity of zero. The mathematical model for this situation 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 Rectangular Coordinates

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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The solution of $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 0 ) = 0 , R ( 1 ) = 0$ is

Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary. The mathematical model of this situation is

In changing from Cartesian to polar coordinates, $\frac { \partial u } { \partial x }$ is

In the previous problem, after separating variables using $u ( r , t ) = R ( r ) T ( t )$ . the resulting problems are

Consider the steady-state temperature distribution in a circular cylinder of radius 2 and height 3, with zero temperature at $r = 2$ and at $z = 3$ and a temperature of 10 at $z = 0$ . The mathematical model for this problem is

In the previous problem, separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems are

In changing from Cartesian to spherical coordinates, $\frac { \partial r } { \partial x }$ becomes

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of $1 - \sin ( \pi r / 2 )$ and an initial velocity of zero. The mathematical model for this situation is