Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / A / 4 / ii

At order one, $({\partial }_t^2+1)y_0=0$, soAt order $ϵ$,The operator sends a constant to itself and sends $\cos 2t,\sin 2t$ to minus three times themselves. HenceAn additional homogeneous $\cos t,\sin t$ contribution can be absorbed into the definitions of $A,B$. There is no resonance forcing at this order; the first amplitude evolution therefore occurs on the chosen slower scale $T={ϵ}^{2}t$.