A Beam of Length $L$ Is Simply Supported at One End and Free at the End

A beam of length $L$ is simply supported at one end and free at the other end. The weight density is constant, $\omega ( x ) = \omega _ { 0 }$ . Let $y ( x )$ represent the deflection at point $x$ . The correct form of the boundary value problem for this beam is

A) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
B) $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } / E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
C) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \omega _ { 0 } E I , y ( 0 ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
D) $\frac { d ^ { 4 } y } { d x ^ { 4 } } = \omega _ { 0 } E I , y ( 0 ) = 0 , y ^ { \prime \prime } ( 0 ) = 0 , y ^ { \prime \prime } ( L ) = 0 , y ^ { \prime \prime \prime } ( L ) = 0$
E) none of the above