Suppose a Spring Has Mass $M$ And Spring Constant $k$ And Let $\omega = \sqrt { k / M }$

Suppose a spring has mass $M$ and spring constant $k$ and let $\omega = \sqrt { k / M }$ . Suppose that the damping constant is so small that the damping force is negligible. If an external force $F ( t ) = 4 F _ { 0 } \cos ( \omega t )$ is applied (the applied frequency equals the natural frequency) , use the method of undetermined coefficients to find the equation that describes the motion of the mass.

A) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } e ^ { - \omega t } } { 2 M \omega }$
B) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { 2 F _ { 0 } t } { M \omega } \sin ( \omega t )$
C) $x ( t ) = c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) + \frac { F _ { 0 } t ^ { - 2 } } { 2 M \omega } \sin ( \omega t )$
D) $x ( t ) = \frac { F _ { 0 } t } { 2 M \omega } \left( c _ { 1 } \cos ( \omega t ) + c _ { 2 } \sin ( \omega t ) \right)$
E) $x ( t ) = \frac { F _ { 0 } t ^ { 2 } } { 2 M \omega } \cos ( \omega t )$