I need to write Van der Pol's equation with parameter $0.1$, which is \begin{align*} y''-(0.1)(1-y^{2})y'+y=0\qquad y(0)=1, \qquad y'(0)=0 \end{align*} as a system of ODE's.
I write $y'=x$, and $x'=y''=(0.1)(1-y^{2})x-y$. Thus I obtain \begin{align*} &y'=x\\ &x'=(0.1)(1-y^{2})x-y \end{align*}
The initial values are $x(0)=y'(0)=0$ and $y(0)=1$.
Can someone check if my system and initial values are correct? Thank you.
Yes, that is correct. For larger parameters than $\mu=0.1$ the numerical simulation works better, that is, allows larger step sizes, when using the fast-slow parametrization $$ \mu v = y' - \mu(1-\tfrac13y^2)y, $$ so that \begin{align} y'&=\mu[v+(1-\tfrac13y^2)y], \\ v'&=-\frac{y}\mu. \end{align}