Consider the Problem of a Vibrating String, Tightly-Stretched Between $x = 0$

Consider the problem of a vibrating string, tightly-stretched between $x = 0$ and $x = 1$ , with a fixed initial position, $f ( x )$ , and zero initial velocity. The mathematical problem for the deflection, $u ( x , t )$ , is $( \text { with } u ( 0 , t ) = 0 , u ( 1 , t ) = 0 )$

A) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( x , 0 ) = f ( x ) , u _ { t } ( x , 0 ) = 0$
B) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0 , u ( x , 0 ) = f ( x ) , u _ { t } ( x , 0 ) = 0$
C) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( x , 0 ) = f ( x ) , u _ { t } ( x , 0 ) = 0$
D) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( x , 0 ) = f ( x ) , u _ { t } ( x , 0 ) = 0$
E) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( x , 0 ) = 0 , u _ { t } ( x , 0 ) = f ( x )$