differential equations - harmonic motion

[sugarplum31]

1. The problem statement, all variables and given/known data

A mass of 5 kilograms dangles from a spring, stretching the spring centimeters when in equilibrium.
The mass oscillates vertically in a fluid, with a frictional force times its speed.
If initially it starts from rest at an extension below the equilibrium point of centimeters, describe the subsequent motion, with a plot of the extension against time and with a phase space plot.


2. Relevant equations

I looked through my book, but my first problem is that I can't figure out if it is Simple Harmonic Motion, or Forced Harmonic Motion. I don't see any specific equations for this that involves all of the variables I have.

3. The attempt at a solution

[Shooting Star]

Write the eqn of motion first. Take the downward direction as positive.

The weight mg acts downward, the buoyant force B acts upward, the frictional force bv acts opp to the velo, where b is a constant, and k is the spring constant, x is the displacement from the equilibrium posn.

ma = mg - B - bv - kx =>
md^2x/dt^2 = mg - B - bdx/dt - kx.

Sub in the values of the constants. This is damped harmonic oscillation.