This is ODE with constant coefficients..
Suppose that the solution is y=e^rx, than solve it!
i find that r=2, r=-2, r=i, r=-i
which means that
complementary solution of eqn is y=c_1*e^2x + c_2*e^-2x + c_3(sinx) + c_4(cosx)
Think about you are in the space with no gravity. When you pull the one end of ruller you will create torque on it. Because of this free end rotates with respect to center of mass.
Then you can think about the gravity. Gravity creates oscillation here?
Is it clear?