I have solved the unforced van der Pol oscillator,
$$\frac{\mathrm d^2x}{\mathrm dt^2} + \epsilon (x^2-1)\frac{\mathrm dx}{\mathrm dt}+\omega_0^2x=0$$
using the multiple time scaling method up to $O(\epsilon)$.
$$x(t,\epsilon)=X(t,\tau,\epsilon)=X_0(t,\tau)+\epsilon\,X_1(t,\tau)+\ldots$$
But I don't know how to work out the solutions using the initial conditions, $x(0,\epsilon)=1$ and $\frac{\mathrm dx(0,\epsilon)}{\mathrm dt}$=0
I can't seem to work out how $(1)$ implies $(2)$ and $(3)$.
$$X(0,0,\epsilon)=1, \frac{\partial X(0,0,\epsilon)}{\partial t}+\epsilon \frac{\partial X(0,0,\epsilon)}{\partial\tau}=0\tag1$$
$$\implies X_0(0,0)=1, \frac{\partial X_0(0,0)}{\partial t}=0\tag2$$
$$X_1(0,0)=0, \frac{\partial X_1(0,0)}{\partial t}+\frac{\partial X_0(0,0)}{\partial \tau}=0\tag3$$