Mathieu's equation, defined as, đều gt2 + [w2 + 2e cos(2t)] u = 0, describes the stability of a pendulum with natural frequency w whose pivot point is oscillated in the vertical direction at a frequency of 2 and amplitude of 2€. a) assume a regular perturbation expansion, u= u(t) + eui(t) + and solve for uo, uz if the initial conditions are u(0) = 1 and du (0) = 0. b) what modes are resonant at order ? c) write down the equation for order o(42) and determine what frequencies are resonant at the next order.