Averaging method (parameter variation) Use the averaging method (parameter variation) to determine a uniform first-order approximation for:
$$ y''+y+ \epsilon y'^5=0$$
for $\epsilon\ll1$.
First transform into polar coordinates, $y=r\sin(\phi)$, $y'=r\cos\phi$. Then compute the derivatives in polar coordinates
\begin{align}
rr'&amp;=y'(y''+y)=-ϵy'^6=-ϵr^6\cos^6\phi
\\
r^2\phi'&amp;=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'&amp;=-ϵ\bar r^5·\frac{20}{64}
\\
\bar \phi'&amp;=1
\end{align}
so that $\bar \phi(t)=t+\phi_0$ and
$$
\bar r(t)=\frac{r(0)}{\sqrt[4]{1+\frac54ϵr(0)^4t}}.
$$
First transform into polar coordinates, $y=r\sin(\phi)$, $y'=r\cos\phi$. Then compute the derivatives in polar coordinates
\begin{align}
rr'&amp;=y'(y''+y)=-ϵy'^6=-ϵr^6\cos^6\phi
\\
r^2\phi'&amp;=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'&amp;=-ϵ\bar r^5·\frac{20}{64}
\\
\bar \phi'&amp;=1
\end{align}
so that $\bar \phi(t)=t+\phi_0$ and
$$
\bar r(t)=\frac{r(0)}{\sqrt[4]{1+\frac54ϵr(0)^4t}}.
$$