I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation:
\begin{equation} \frac{d^2y}{dt^2} + y = \epsilon \bigg(\frac{dy}{dt} - \frac{1}{3}\bigg(\frac{dy}{dt}\bigg)^3\bigg). \end{equation}
If I plug in the perturbation series
\begin{equation} y = y_0 + \epsilon y_1 + \mathcal{O}(\epsilon^2), \end{equation}
and solve for the $y_n$ order by order, I get $y_0 = Ae^{it}$ + complex conjugate, and, courtesy of Mathematica,
\begin{equation} y_1 = \alpha e^{it} + \beta t e^{it} + \gamma e^{3it} + \text{complex conjugate}, \end{equation} where \begin{equation} \alpha = \frac{iA}{4}\big(1 - A\bar{A}) + C, \end{equation} where $C$ is the (complex) constant of integration specific to $y_1$. The other two constants are \begin{equation} \beta = \frac{A}{2}\big(1 - A\bar{A}\big), \end{equation} and \begin{equation} \gamma = -\frac{iA^3}{24}. \end{equation}
PROBLEM: I'm having difficulty matching boundary conditions. Say my initial conditions are $y(0) = y0$ and $dy/dt(0) = \text{v}0$. I believe that in order for my series solution to be valid for all $\epsilon$, I must have $y_0(0) = y0$, $dy_0/dt(0) = \text{v}0$, and $y_n(0) = 0$ and $dy_n/dt(0) = 0$ for all $n \ge 1$.
QUESTION 1: Is that true in general for perturbation series? i.e. that $y_n(0) = 0$ and $dy_n/dt(0) = 0$ for all $n \ge 1$?
QUESTION 2: When I go to match $y_1(0) = 0$, I seem to require that $\alpha + \gamma = 0$, so $C$ should be whatever makes $\alpha = iA^3/24$. But this seems to mean that I have used my first boundary condition ($y_1(0) = 0$) to determine both the real and imaginary parts of $C$. In particular, I have not used the condition $dy_1/dt(0) = 0$ to help set C, and as one might expect, the resulting solution \begin{equation} y_1 = \frac{iA^3}{24}e^{it} + \frac{A}{2}\big(1 - A\bar{A}\big)te^{it} - \frac{iA^3}{24}e^{3it} + \text{complex conjugate} \end{equation} yields $dy_1/dt(0) \ne 0$. Where's the catch? I have some reason to believe this is the right expression for $y_1$ since a published research article uses this form, but I don't understand how it's possible for $dy_1/dt(0)$ to not equal 0. Any insight would be greatly appreciated.