In the Previous Two Problems, the Solution For $u ( x , y )$

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

A) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi x ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$
B) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \sinh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$
C) $u = c _ { 0 } y + \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \sinh ( n \pi y )$ , where $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) d x / 4$ and
$$c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$$
D) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \cos ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$
E) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin ( n \pi x ) \cosh ( n \pi y ) , \text { where } c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) \cos ( n \pi x ) d x / \sinh ( 2 n \pi )$