Is my forced spring-mass system correctly modeled with these equations?

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The discussion centers on modeling a forced spring-mass system using differential equations. The user seeks validation for their equations that account for gravitational force and a forcing term, contrasting them with a simple homogeneous equation represented by x = A cos(w0t). The key focus is on ensuring the correct representation of forces acting on the system after an external force is applied and released.

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After the spring-mass is in equilibrium, a force is applied then released at t0 and the spring-mass system is set into motion. For a certain period of time the spring-mass will move around the x0 equilibrium position until the restoring from the spring has balanced the gravitational force.

Are my two equations correct? I guess I'm having trouble accounting for the gravitational force and the forcing term in the DE. Otherwise, it would be a simple homogeneous equation with x = A cos(w0t).

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-09-21193429.jpg?t=1285115763

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-09-21193118.jpg?t=1285115676
 
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