First, I'm a second year high school physics student so my thinking may be half-baked. I was wondering what motion a piston would exhibit if it was the frictionless lid of an ideal gas cylinder with a constant amount of air, and if it was pushed or pulled a little bit. There would be a restoring force exerted by the atmospheric pressure - pressure inside the cylinder. At first I thought it would simple harmonic motion, but then I visualized a real-world experiment and thought that the motion would be more like y = 1 - e^-x. I set f = ma = m*(second derivative of y(t)) equal to the restoring force, which is a constant times (1/ h+y) where h is the equilibrium height of the piston. I don't know how to solve second order differential equations. I reasoned that the second derivative of y(t) is equal to a function of the form 1/y. Therefore y(t) is of the form of the second antiderivative of y^-1, which is the integral of ln(x), which is xln(x)-x. The graph of xln(x)-x looks nothing like the motion the piston should exhibit. Could you help me understand this?