How is the error in the expanded equation (9) identified and resolved?

[Forhad3097]

In this articlehttp://arxiv.org/abs/0802.3525, the authors present the inhomogeneous equation

$$\ddot{\phi}_2 + \phi_2 + g_2\phi_1^2 + \omega_1\ddot{\phi}_1 = 0\tag{11}$$

followed by its solution

$$\phi_2 = p_2\cos(\tau + \alpha) + q_2\sin(\tau + \alpha) + \frac{g_2}{6}p_1^2[\cos(2\tau + 2\alpha) - 3] + \frac{\omega_1}{4}p_1[2\tau\sin(\tau + \alpha) + \cos(\tau + \alpha)]\tag{14}$$

How is this solution obtained?

 

[Bill_K]

You left out a crucial part of the solution: φ₁ = p₁ cos(τ + α). Plug that into the equation and you have a driven harmonic oscillator, solved in any calc book.

 

[Forhad3097]

Find out the error please

I have expanded the equation(9) from the paper http://arxiv.org/abs/0802.3525 . $$-[1+\epsilon \omega_1+ \epsilon^2 \omega_2+\epsilon^3 \omega_4...][\epsilon \ddot\phi_1+\epsilon^2 \ddot\phi_2+\epsilon^3 \ddot\phi_3...]+\epsilon^2[\epsilon \Delta\phi_1 + \epsilon^2 \Delta\phi_2+\epsilon^3 \Delta\phi_3...]=\epsilon \phi_1+\epsilon^2 \phi_2+\epsilon^3 \phi_3...+[g_2 \phi^2 + g_3 \phi^3...]$$
From this equation We can write equation (10), (11) and (12) and so on. but equation (12) is not matching with the expanding equation I did. I think problem in expanding with $$[g_2 \phi^2 + g_3 \phi^3...]$$.
can you please have a look, that where is the problem?