Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2010 / A / 3 / i

Matching powers of $ϵ$ gives $u_0^{\prime \prime }+u_0=0$, $u_1^{\prime \prime }+u_1=u_0^2$, and $u_2^{\prime \prime }+u_2=2u_0{u}_1$, with $u_0(0)=2$ and homogeneous initial data for every higher correction. ThusThere is no resonant fundamental forcing at first order. At second order,The first term resonates with the homogeneous oscillator. Solving with the initial data givesConsequently the regular perturbation approximation is $u=u_0+ϵu_1+{ϵ}^2{u}_2+O({ϵ}^3)$ for fixed $t$. The first secular term is $(10/3){ϵ}^{2}t\sin t$, at order ${ϵ}^2$. It makes the regular expansion nonuniform on times of order ${ϵ}^{-2}$.