2nd order ODE including a quadratic
I am trying to solve for the rocking frequency of a semi-circular object (imagine a cylinder cut down the z axis so that the cross section is semi-circular), and I have got stuck right at the end.
I figured out the location of the centre of mass so that I could write down the torque experienced by the object when sitting at any angle, and am now faced by a differential equation that looks substantially trickier than I imagined.
Note that I converted sines and cosines to the following:
sin(x) ---> x
cos(x) ---> 1 - 0.5x^2
After all this work I am left with an equation of the form: x'' = f(x)
Where f(x) contains a linear and quadratic term.
Can anyone give me any clues as to how I might attack this equation?