Mathieu's equation 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?....(see more details in the attachment, thanks)

2. Mathieu's equation, dened as, '12\" + [0.12 + 26 cos(2t)] u = 0 alt2 ' 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 26. a) Assume a regular perturbation expansion, u = u0(t) + eu1(t) + ' ~, and solve for U0, U1 if the initial conditions are u(0) = l and '31:(0) = 0. b) What modes are resonant at order 6? c) Write down the equation for order C(62) and determine what frequencies are resonant at the next order. You do not need to solve this equation