The Differential Equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0$

Multiple Choice

The differential equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0$ is a model for an undamped spring-mass system with a nonlinear restoring force. The initial conditions are $x ( 0 ) = 0$ , $x ^ { \prime } ( 0 ) = 1$ . The solution of the linearized system is

A) $x = \left( e ^ { t } + e ^ { - t } \right) / 2$
B) $x = \left( e ^ { t } - e ^ { - t } \right) / 2$
C) $x = \cos t$
D) $x = \sin t$
E) $x = \cos t - \sin t$