Spring-mass System Phase Plane 
If a spring-mass system has a friction force proportional to the cube of velocity, then
  $$m\frac{d^2x}{dt^2} + a\left(\frac{dx}{dt}\right)^3 + kx = 0$$
(a) Derive a first-order differential equation describing the phase plane ($\frac{dx}{dt}$ as a function of $x$).
(b) Sketch the solution in the phase plane.

For part (a), I let $y=\frac{dx}{dt}$ and eventually reached $m\frac{dy}{dt} + ay^3 + kx= 0$.
Then $\frac{dy}{dx}= \frac{-ay^3 - kx}{my}$. 
I found the equilibrium point to be $(0,0)$.
I don't really know if any of what I solved is correct, but matters got worse for part (b). How would I sketch a solution if I do not have information about $m$, $a$ and $K$?
 A: I don't know how much detail you need, but if you only need to draw a very rough phase-space diagram you can do this using only some very basic physical reasoning.
If $a=0$ then the friction force vanishes and consequently energy (kinetic plus potential energy of the spring) is conserved. The equation system in this case can be written
$$\frac{dE}{dt} = \frac{d}{dt}\left[\frac{1}{2}mx'^2 + \frac{1}{2}kx^2\right] = 0$$
so with $y= \frac{dx}{dt}$ we have $$my^2 + kx^2 = C$$
which is the equation for an ellipse in the $(x,y)$ phase-space.
Now when introducing friction, $a\not=0$, the energy will be drained from the system as we go along and the system will exibit damped osccilations. This can also be seen from
$$\frac{dE}{dt} = \frac{d}{dt}\left[\frac{1}{2}mx'^2 + \frac{1}{2}kx^2\right] = -a\left(\frac{dx}{dt}\right)^4 < 0$$
This implies that the phase-space path of the system is something that looks like a spiral that goes inwards towards $(x,y) = (0,0)$.
If you want to derive a more detailed path then try studying the system when the friction term dominates compared to when it is subdominant (compared to the potential energy of the spring). Note that the latter case (which is what happens close to the fixpoint is covered by the case $a=0$ studied above).
If you have the appropriate software it is fairly simple to do this numerically to check how the true solution looks like. Below is a numerical solution showing a typical example of the $(x,y)$ phase-space:
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(source: folk.uio.no)
