Question: Find the leading order of the asymptotic expansion for large t: $$ \frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=F\cos\Bigl(\frac{1}{3}\bigl(1+\varepsilon\omega\bigr)t\Bigr) $$
I have tried to solve the problem via regular multiple scale analysis by transforming $t_0=t$, $t_1=\varepsilon t$ & expressing the solution in the form of an asymptotic expansion $$x(t)=y(t_0,t_1)=\sum_{n=0}^\infty \varepsilon^ny_n(t_0,t_1).$$
The leading order problem becomes: $$ \frac{\partial^2y_0}{\partial t_0}+y_0=F\cos\Bigl(\frac{1}{3}(t_0+\omega t_1)\Bigr) $$ solving, i obtain $$y_0=R_0(t_1)\cos(t_0+\phi_0(t_1))+\frac{9}{8}\cos\Bigl(\frac{1}{3}(t_0+\omega t_1)\Bigr)$$
After solving the leading order the 1st order problem is as follows $$\small\begin{align} \frac{\partial^2 y_1}{\partial t_0}+y_1 &=\sin(t_0+\phi_0)(2R_0'+\beta R_0)+\cos(t_0+\phi_0)(2R_0\phi_0') \\&~~~+\frac{\omega}{4}F\cos(\frac{1}{3}(t_0+\omega t_1)) +\frac{3}{8}\beta F\sin(\frac{1}{3}(t_0+\omega t_1)) \\&~~~-R_0^3\left(\frac{1}{4}\cos(3(t_0+\phi_0))) +\frac{3}{4}\cos((t_0+\omega t_1))\right) \\&~~~-(\frac{9}{8}F)^3(\frac{1}{4}\cos(t_0+\omega t_1) +\frac{3}{4}\cos(\frac{1}{3}(t_0+\omega t_1))) \\&~~~-3R_0(\frac{9}{8}F)^2\left(\frac{1}{4}\cos\big(\frac{1}{3}t_0+\phi_0-\frac{2}{3}(\omega t_1)\big) +\frac{1}{4}\cos(\frac{4}{3}t_0+\phi_i+\frac{2}{3}\omega t_1) +\frac{1}{2}\cos(t_0 +\phi_0)\right) \\&~~~-3R_0^2(\frac{9}{8}F)\left(\frac{1}{4}\cos(\frac{5}{2}t_0+2\phi_0+\frac{1}{3}\omega t_1)+\frac{1}{4}\cos(\frac{3}{2}t_0+2\phi_0-\frac{1}{3}\omega t_1)+\frac{1}{2}\cos(\frac{1}{3}(t_0+\omega_1 t_1))\right) \end{align}$$ demanding the coefficent of the $\sin(t_0+\phi_0)$ term to vanish (for large $t$) i obtain $R_0=0$, thus the equation becomes $$\small\begin{align} \frac{\partial^2 y_1}{\partial t_0}+y_1 &=\frac{\omega}{4}F\cos(\frac{1}{3}(t_0+\omega t_1))+\frac{3}{8}\beta F\sin(\frac{1}{3}(t_0+\omega t_1)) \\&~~~-(\frac{9}{8}F)^3\left(\frac{1}{4}\cos(t_0+\omega t_1)+\frac{3}{4}\cos(\frac{1}{3}(t_0+\omega t_1)))\right) \end{align}$$ the problem is that a secular term remains (i.e. $-(\frac{9}{8}F)^3(\frac{1}{4}\cos(t_0+\omega t_1))$)?