Consider the following problem:
$$\frac{\text{d}^2y}{\text{d}t^2} + y + \epsilon\ y^3 = 0\ \ \ \ \text{for}\ \ t \geq 0, y(0) = A, y'(0)=0$$
Compute an approximate solution by substitution method:
$y(\tau) = y_0(\tau) + \epsilon\ y_1(\tau) + \epsilon^2y_2(\tau) + \ldots$
$\tau = w(\epsilon)t = (1+\epsilon w_1 + \epsilon^2 w_2 + \ldots)t$
$\text{up to the order}\ \epsilon^2$;
I have hard time using such substitution / perturbation method to approximate the exact solution. Any help?
Set $y(t)=r(t)\cos(w(t))$ and $\dot y(t)=-r(t)\sin(w(t))$. Then compute $$ \dot r=\frac{d}{dt}\sqrt{y^2+\dot y^2} $$ and $$ \dot w=-\frac{d}{dt}\arctan \frac{\dot y}{y} $$ and insert the differential equation to remove second order derivatives and then the zero-order solutions to obtain the first order approximations.
Another approach is to observe that the system is conservative with first integral $$ E(\dot y, y)=\dot y^2+y^2+\fracϵ2y^4, \quad E(0,a)=a^2+\fracϵ2a^4 $$ so that the solutions are periodic and symmetric. One quarter of the period length $2\pi w(ϵ)$ can be obtained by integrating $$ w(ϵ)\frac\pi2=\int_0^a\frac1{\sqrt{(a^2-y^2)(1+\fracϵ2(a^2+y^2))}}dy \\=\int_0^{\pi/2}\frac1{\sqrt{1+\frac{ϵa^2}2(1+\sin^2u)}}du $$