My question is:
Apply the regular perturbation theory ansatz to solve the ivp
$$x''(t) + x(t) + \epsilon{x^2(t)} = 0$$ $$∀t > 0$$
subject to $x(0) = 0$ and $x'(0) = 1$ to first order in $\epsilon$
Im not 100% confident in executing these types of questions so if anyone could help then that would be great.
In first orders in $ϵ$ you get the perturbed solution $x=x_0+ϵx_1+ϵ^2x_2+O(ϵ^3)$. Inserting into the equation and separating powers of $ϵ$ you get the system \begin{align} \ddot x_0+x_0&=0 \\ \ddot x_1+x_1&=-x_0^2 \\ \ddot x_2+x_2&=-2x_0x_1 \\ \end{align}
With the initial conditions the first equation has the solution $$x_0(t)=\sin t.$$ As $$-x_0^2=-\sin^2t=-\frac12(1-\cos(2t)),$$ the second equation has a particular solution $$x_1(t)=-\frac12(1-\cos(t))-\frac16(\cos(2t)-\cos(t)) \\ =-\frac13(1-\cos t)^2=-\frac43\sin^4\Bigl(\frac t2\Bigr) $$ where homogeneous solution terms were added to get value and derivative zero in each term.
And so forth. Resonance terms that will turn up can be an indication of a frequency shift, reversing the Taylor expansion one can contract $\sin t+\delta t\cos t=\sin((1+δ)t)+O(δ^2)$, $δ$ some equally small expression in $ϵ$.