Oscillations of a Piston in a cylinder containing a trapped gas

[Varak]

A frictionless cylinder of cross-sectional area A contains a gas that is trapped by a piston of mass m that fits the cylinder tightly but is free to move up and down.

It is open to atmospheric pressure (PA) on one end.

The piston is slightly displaced and when released oscillates about its equilibrium position.

Find the frequency of small oscillations and show that the oscillations are approximately harmonic with frequency w^2=gamma(g/H+P0A/mH), where H is the height of the trapped gas.

The hint is that it is an adiabatic compression where PV^gamma = constant and gamma=5/3. We're supposed to write P(y) and expand in a power series.
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Here is what I have so far.

The three forces on the piston are -mg - PA + P(y)A

P0 = Initial Pressure
V0 = Initial Volume = HA

P0*V0^gamma = P(y)*V(y)^gamma

P0*HA^gamma = P(y)*((H+y)A)^gamma

P(y)=(P0*H^gamma)/(H+y)^gamma

I'm guessing this is where I need to expand into a power series, but I'm not sure how to do so. Any help would be appreciated.

Also, if someone could tell me how to input equations in a better format, I'd appreciate it. I've seen many posts with Mathcad-like text which is much easier to read, but, since I'm a new user, I don't know how to input all the special characters.

 

[sic]

yes, you take the first order approximation:
dP(y)/dy=Whatever
hence
dP(y)=Whatever*dy
F=dP(y)*A=Whatever*A*dy (harmonic oscillator equation)
and k/m=omega^2=Whatever*A/m
now pay attention that
PO=PA+mg/A