Motion of Cart attached to Wall by Spring

Problem Definition
Then the motion of $C$ is described by the second order ODE:
 * $\dfrac {\d^2 \mathbf x} {\d t^2} + \dfrac k m \mathbf x = 0$

Proof
By Newton's Second Law of Motion, the force on $C$ equals its mass times its acceleration:


 * $\mathbf F = m \mathbf a$

By Acceleration is Second Derivative of Displacement with respect to Time:
 * $\mathbf a = \dfrac {\d^2 \mathbf x} {\d t^2}$

By Hooke's Law:
 * $\mathbf F = -k \mathbf x$

So: