Averaging method (parameter variation)

Question

Use the averaging method (parameter variation) to determine a uniform first-order approximation for: $$ y''+y+ \epsilon y'^5=0$$ for $\epsilon\ll1$.

Answer 1

First transform into polar coordinates, $y=r\sin(\phi)$, $y'=r\cos\phi$. Then compute the derivatives in polar coordinates \begin{align} rr'&=y'(y''+y)=-ϵy'^6=-ϵr^6\cos^6\phi \\ r^2\phi'&=y'^2-yy''=r^2-y(y+y'')=r^2-ϵr^6\cos^5\phi\sin\phi \end{align} Now take the average as $\phi$ varies over $[0,2\pi]$ \begin{align} \bar r'&=-ϵ\bar r^5·\frac{20}{64} \\ \bar \phi'&=1 \end{align} so that $\bar \phi(t)=t+\phi_0$ and $$ \bar r(t)=\frac{r(0)}{\sqrt[4]{1+\frac54ϵr(0)^4t}}. $$