3.

In this problem, we will make a small excursion into the nonlinear world for second-order

systems. Don't worry that we have not yet covered the material in class.

Consider an approximate version of the equation governing the motion of a harmonically excited

pendulum:

x+(x-x³/6)= F cos(at),

and explore the possibility of finding an 'approximate' periodic (really only harmonic) solution at the

excitation frequency o by assuming the solution to be

x(t) = Acos(at) + B sin(cot),

where A and B are constants to be determined. Substitute this solution form in the differential equation,

expand the functions of (@t) in a Fourier series, and then compare coefficients of cos(ot) and sin(cot)

on the two sides of the resulting expression. Note that this gives two simultaneous nonlinear algebraic

equations in the two unknowns A and B. Their solutions provide 'approximations' to the possible

periodic solutions of the appropriate type!!. See if you can solve the two algebraic equations.