Current rules I am using in simplifications are
Verify the above are valid for \(x\rightarrow 0^{+}\) and well for \(x\rightarrow \infty \). What can we say about \(\left ( S^{\prime }\right ) \) compared to \(\left ( S^{\prime }\right ) ^{2}\)?
\[ y^{\prime \prime }=\frac{1}{x^{5}}y \] Irregular singular point at \(x\rightarrow 0^{+}\). Let \(y=e^{S_{0}\left ( x\right ) }\) and the above becomes
\begin{align*} y\left ( x\right ) & =e^{S_{0}\left ( x\right ) }\\ y^{\prime }\left ( x\right ) & =S_{0}^{\prime }e^{S}\\ y^{\prime \prime } & =S_{0}^{\prime \prime }e^{S_{0}}+\left ( S_{0}^{\prime }\right ) ^{2}e^{S_{0}}\\ & =\left ( S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2}\right ) e^{S_{0}} \end{align*}
Substituting back into \(\frac{d^{2}}{dx^{2}}y=x^{-5}y\) gives\begin{align*} \left ( S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2}\right ) e^{S} & =x^{-5}e^{S_{0}\left ( x\right ) }\\ S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2} & =x^{-5} \end{align*}
Before solving for \(S_{0}\), we can do one more simplification. Using the approximation that \(\left ( S_{0}^{\prime }\right ) ^{2}\ggg S_{0}^{\prime \prime }\) for \(x\rightarrow x_{0}\), the above becomes\[ \left ( S_{0}^{\prime }\right ) ^{2}\thicksim x^{-5}\] Now we are ready to solve for \(S_{0}\)\begin{align*} S_{0}^{\prime } & \thicksim \omega x^{-\frac{5}{2}}\\ S_{0} & \thicksim \omega \int x^{-\frac{5}{2}}dx\\ & \thicksim \omega \frac{x^{-\frac{3}{2}}}{-\frac{3}{2}}\\ & \thicksim -\frac{2}{3}\omega x^{-\frac{3}{2}} \end{align*}
To find leading behavior, let \[ S\left ( x\right ) =S_{0}\left ( x\right ) +S_{1}\left ( x\right ) \] Then \(y\left ( x\right ) =e^{S_{0}\left ( x\right ) +S_{1}\left ( x\right ) }\) and hence now\begin{align*} y^{\prime }\left ( x\right ) & =\left ( S_{0}\left ( x\right ) +S_{1}\left ( x\right ) \right ) ^{\prime }e^{S_{0}+S_{1}}\\ y^{\prime \prime }\left ( x\right ) & =\left ( \left ( S_{0}+S_{1}\right ) ^{\prime }\right ) ^{2}e^{S_{0}+S_{1}}+\left ( S_{0}+S_{1}\right ) ^{\prime \prime }e^{S_{0}+S_{1}} \end{align*}
Using the above, the ODE \(\frac{d^{2}}{dx^{2}}y=x^{-5}y\) now becomes\begin{align*} \left ( \left ( S_{0}+S_{1}\right ) ^{\prime }\right ) ^{2}e^{S_{0}+S_{1}}+\left ( S_{0}+S_{1}\right ) ^{\prime \prime }e^{S_{0}+S_{1}} & \thicksim x^{-5}e^{S_{0}+S_{1}}\\ \left ( \left ( S_{0}+S_{1}\right ) ^{\prime }\right ) ^{2}+\left ( S_{0}+S_{1}\right ) ^{\prime \prime } & \thicksim x^{-5}\\ \left ( S_{0}^{\prime }+S_{1}^{\prime }\right ) ^{2}+S_{0}^{\prime \prime }+S_{1}^{\prime \prime } & \thicksim x^{-5}\\ \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime } & \thicksim x^{-5} \end{align*}
But \(S_{0}^{\prime }=\omega x^{-\frac{5}{2}}\), found before, hence \(\left ( S_{0}^{\prime }\right ) ^{2}=x^{-5}\) and the above simplifies to\[ \left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }=0 \] Using approximation \(S_{0}^{\prime }S_{1}^{\prime }\ggg \left ( S_{1}^{\prime }\right ) ^{2}\) the above simplifies to\[ 2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }=0 \] Finally, using approximation \(S_{0}^{\prime \prime }\ggg S_{1}^{\prime \prime },\) the above becomes\begin{align*} 2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime } & =0\\ S_{1}^{\prime } & \thicksim -\frac{S_{0}^{\prime \prime }}{2S_{0}^{\prime }}\\ S_{1} & \thicksim -\frac{1}{2}\ln S_{0}^{\prime }+c\\ S_{1}^{\prime } & \thicksim -\frac{1}{2}\ln x^{-\frac{5}{2}}+c\\ S_{1}^{\prime } & \thicksim \frac{5}{4}\ln x+c \end{align*}
Hence, the leading behavior is\begin{align} y\left ( x\right ) & =e^{S_{0}\left ( x\right ) +S_{1}\left ( x\right ) }\nonumber \\ & =\exp \left ( -\frac{2}{3}\omega x^{-\frac{3}{2}}+\frac{5}{4}\ln x+c\right ) \nonumber \\ & =cx^{\frac{5}{4}}\exp \left ( -\omega \frac{2}{3}x^{-\frac{3}{2}}\right ) \tag{1} \end{align}
To verify, using the formula 3.4.28, which is\[ y\left ( x\right ) \thicksim cQ^{\frac{1-n}{2n}}\exp \left ( \omega \int ^{x}Q\left ( t\right ) ^{\frac{1}{n}}dt\right ) \] In this case, \(n=2\), since the ODE \(y^{\prime \prime }=x^{-5}y\) is second order. Here we have \(Q\left ( x\right ) =x^{-5}\), therefore, plug-in into the above gives \begin{align} y\left ( x\right ) & \thicksim c\left ( x^{-5}\right ) ^{\frac{1-2}{4}}\exp \left ( \omega \int ^{x}\left ( t^{-5}\right ) ^{\frac{1}{2}}dt\right ) \nonumber \\ & \thicksim c\left ( x^{-5}\right ) ^{\frac{-1}{4}}\exp \left ( \omega \int ^{x}t^{-\frac{5}{2}}dt\right ) \nonumber \\ & \thicksim cx^{\frac{5}{4}}\exp \left ( \omega \left ( \frac{x^{-\frac{3}{2}}}{-\frac{3}{2}}\right ) \right ) \nonumber \\ & \thicksim cx^{\frac{5}{4}}\exp \left ( -\omega \frac{2}{3}x^{-\frac{3}{2}}\right ) \tag{2} \end{align}
Comparing (1) and (2), we see they are the same.
\[ y^{\prime \prime \prime }=xy \] Irregular singular point at \(x\rightarrow +\infty \). Let \(y=e^{S_{0}\left ( x\right ) }\) and the above becomes\begin{align*} y\left ( x\right ) & =e^{S_{0}\left ( x\right ) }\\ y^{\prime }\left ( x\right ) & =S_{0}^{\prime }e^{S_{0}}\\ y^{\prime \prime } & =S_{0}^{\prime \prime }e^{S_{0}}+\left ( S_{0}^{\prime }\right ) ^{2}e^{S_{0}}\\ & =\left ( S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2}\right ) e^{S_{0}}\\ y^{\prime \prime \prime } & =\left ( S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2}\right ) ^{\prime }e^{S_{0}}+\left ( S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{2}\right ) S_{0}^{\prime }e^{S_{0}}\\ & =\left ( S_{0}^{\prime \prime \prime }+2S_{0}^{\prime }S_{0}^{\prime \prime }\right ) e^{S_{0}}+\left ( S_{0}^{\prime }S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3}\right ) e^{S_{0}}\\ & =\left ( S_{0}^{\prime \prime \prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3}\right ) e^{S_{0}} \end{align*}
Substituting back into \(y^{\prime \prime \prime }=xy\) gives\begin{align*} \left ( S_{0}^{\prime \prime \prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3}\right ) e^{S_{0}} & =xe^{S_{0}\left ( x\right ) }\\ S_{0}^{\prime \prime \prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3} & =x \end{align*}
Before solving for \(S_{0}\), we can do one more simplification. Using the approximation that \(\left ( S_{0}^{\prime }\right ) ^{3}\ggg S_{0}^{\prime \prime \prime }\) for \(x\rightarrow x_{0}\), the above becomes\[ 3S_{0}^{\prime }S_{0}^{\prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3}\thicksim x \] In addition, since \(S_{0}^{\prime }\ggg S_{0}^{\prime \prime }\) then we can use the approximation \(\left ( S_{0}^{\prime }\right ) ^{3}\ggg S_{0}^{\prime }S_{0}^{\prime \prime }\) and the above becomes\begin{align*} \left ( S_{0}^{\prime }\right ) ^{3} & \thicksim x\\ S_{0}^{\prime } & \thicksim \omega x^{\frac{1}{3}}\\ S_{0} & \thicksim \omega \int x^{\frac{1}{3}}dx\\ S_{0} & \thicksim \omega \frac{3}{4}x^{\frac{4}{3}} \end{align*}
To find leading behavior, let \[ S\left ( x\right ) =S_{0}\left ( x\right ) +S_{1}\left ( x\right ) \] Then \(y\left ( x\right ) =e^{S_{0}\left ( x\right ) +S_{1}\left ( x\right ) }\) and hence now\begin{align*} y^{\prime }\left ( x\right ) & =\left ( S_{0}\left ( x\right ) +S_{1}\left ( x\right ) \right ) ^{\prime }e^{S_{0}+S_{1}}\\ y^{\prime \prime }\left ( x\right ) & =\left ( \left ( S_{0}+S_{1}\right ) ^{\prime }\right ) ^{2}e^{S_{0}+S_{1}}+\left ( S_{0}+S_{1}\right ) ^{\prime \prime }e^{S_{0}+S_{1}}\\ & =\left ( S_{0}^{\prime }+S_{1}^{\prime }\right ) ^{2}e^{S_{0}+S_{1}}+\left ( S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) e^{S_{0}+S_{1}}\\ & =\left ( \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }\right ) e^{S_{0}+S_{1}}+\left ( S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) e^{S_{0}+S_{1}}\\ & =\left ( \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) e^{S_{0}+S_{1}} \end{align*}
We can take the third derivative\begin{align*} y^{\prime \prime \prime }\left ( x\right ) & \thicksim \left ( \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) ^{\prime }e^{S_{0}+S_{1}}\\ & +\left ( \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) \left ( S_{0}+S_{1}\right ) ^{\prime }e^{S_{0}+S_{1}}\\ & \\ & \thicksim \left ( 2S_{0}^{\prime }S_{0}^{\prime \prime }+2S_{1}^{\prime }S_{1}^{\prime \prime }+2S_{0}^{\prime \prime }S_{1}^{\prime }+2S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }\right ) e^{S_{0}+S_{1}}\\ & +\left ( \left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{2}+2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }+S_{1}^{\prime \prime }\right ) \left ( S_{0}^{\prime }+S_{1}^{\prime }\right ) e^{S_{0}+S}\\ & \\ & \thicksim \left ( 2S_{0}^{\prime }S_{0}^{\prime \prime }+2S_{1}^{\prime }S_{1}^{\prime \prime }+2S_{0}^{\prime \prime }S_{1}^{\prime }+2S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }\right ) e^{S_{0}+S_{1}}\\ & +\left ( \left ( S_{0}^{\prime }\right ) ^{3}+S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+2\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+S_{0}^{\prime }S_{0}^{\prime \prime }+S_{0}^{\prime }S_{1}^{\prime \prime }\right ) e^{S_{0}+S_{1}}\\ & +\left ( S_{1}^{\prime }\left ( S_{0}^{\prime }\right ) ^{2}+\left ( S_{1}^{\prime }\right ) ^{3}+2S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+S_{1}^{\prime }S_{0}^{\prime \prime }+S_{1}^{\prime }S_{1}^{\prime \prime }\right ) e^{S_{0}+S_{1}}\\ & \\ & \thicksim \left ( 2S_{0}^{\prime }S_{0}^{\prime \prime }+2S_{1}^{\prime }S_{1}^{\prime \prime }+2S_{0}^{\prime \prime }S_{1}^{\prime }+2S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }\right ) e^{S_{0}+S_{1}}\\ & +\left [ \left ( S_{0}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+S_{0}^{\prime }S_{0}^{\prime \prime }+S_{0}^{\prime }S_{1}^{\prime \prime }+\left ( S_{1}^{\prime }\right ) ^{3}+S_{1}^{\prime }S_{0}^{\prime \prime }+S_{1}^{\prime }S_{1}^{\prime \prime }\right ] e^{S_{0}+S_{1}}\\ & \thicksim \left ( 3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }+3S_{0}^{\prime \prime }S_{1}^{\prime }+3S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }+\left ( S_{0}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+\left ( S_{1}^{\prime }\right ) ^{3}\right ) e^{S_{0}+S_{1}}\\ & \thicksim \left ( \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }+3S_{0}^{\prime \prime }S_{1}^{\prime }+3S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }\right ) e^{S_{0}+S_{1}} \end{align*}
Lets go ahead and plug-in this into the ODE\[ \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }+3S_{0}^{\prime \prime }S_{1}^{\prime }+3S_{0}^{\prime }S_{1}^{\prime \prime }+S_{0}^{\prime \prime \prime }+S_{1}^{\prime \prime \prime }\thicksim x \] Now we do some simplification. \(\left ( S_{0}^{\prime }\right ) ^{3}\ggg S_{0}^{\prime \prime \prime }\) and \(\left ( S_{1}^{\prime }\right ) ^{3}\ggg S_{1}^{\prime \prime \prime }\), hence above becomes\[ \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }+3S_{0}^{\prime \prime }S_{1}^{\prime }+3S_{0}^{\prime }S_{1}^{\prime \prime }\thicksim x \] Also, since \(S_{0}^{\prime \prime }\ggg S_{1}^{\prime \prime }\) then \(3S_{0}^{\prime }S_{0}^{\prime \prime }\ggg 3S_{0}^{\prime }S_{1}^{\prime \prime }\)\[ \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }+3S_{0}^{\prime \prime }S_{1}^{\prime }\thicksim x \] Also, since \(\left ( S_{0}^{\prime }\right ) ^{2}\ggg S_{0}^{\prime \prime }\) then \(3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }\ggg 3S_{0}^{\prime \prime }S_{1}^{\prime }\)\[ \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }+3S_{1}^{\prime }S_{1}^{\prime \prime }\thicksim x \] Also since \(S_{1}^{\prime }\ggg S_{1}^{\prime \prime }\) then \(3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }\ggg 3S_{1}^{\prime }S_{1}^{\prime \prime }\)\[ \left ( S_{0}^{\prime }\right ) ^{3}+\left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }\thicksim x \] But \(S_{0}^{\prime }\thicksim x^{\frac{1}{3}}\) hence\(\left ( S_{0}^{\prime }\right ) ^{3}\thicksim x\) and the above simplies to
\[ \left ( S_{1}^{\prime }\right ) ^{3}+3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }=0 \]
Using \(3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}\ggg \left ( S_{1}^{\prime }\right ) ^{3}\) since \(S_{0}^{\prime }\ggg S_{1}^{\prime }\) then
\[ 3S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}+3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }=0 \]
Using \(3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }\ggg S_{0}^{\prime }\left ( S_{1}^{\prime }\right ) ^{2}\) since \(\left ( S_{0}^{\prime }\right ) ^{2}\ggg S_{0}^{\prime }\) then
\[ 3\left ( S_{0}^{\prime }\right ) ^{2}S_{1}^{\prime }+3S_{0}^{\prime }S_{0}^{\prime \prime }=0 \]
No more simplification. We are ready to solve for \(S_{1}\).
\begin{align*} S_{1}^{\prime } & \thicksim \frac{-S_{0}^{\prime }S_{0}^{\prime \prime }}{\left ( S_{0}^{\prime }\right ) ^{2}}\\ & \thicksim \frac{-S_{0}^{\prime \prime }}{S_{0}^{\prime }} \end{align*}
Hence\begin{align*} S_{1} & \thicksim -\int \frac{S_{0}^{\prime \prime }}{S_{0}^{\prime }}dx\\ & \thicksim -\ln S_{0}^{\prime }+c \end{align*}
Since \(S_{0}^{\prime }\thicksim x^{\frac{1}{3}}\)then the above becomes
\begin{align*} S_{1} & \thicksim -\ln x^{\frac{1}{3}}+c\\ S_{1} & \thicksim -\frac{1}{3}\ln x+c \end{align*}
Hence, the leading behavior is
\begin{align} y\left ( x\right ) & =e^{S_{0}\left ( x\right ) +S_{1}\left ( x\right ) }\nonumber \\ & =\exp \left ( \omega \frac{3}{4}x^{\frac{4}{3}}-\frac{1}{3}\ln x+c\right ) \nonumber \\ & =cx^{\frac{-1}{3}}\exp \left ( \omega \frac{3}{4}x^{\frac{4}{3}}\right ) \tag{1} \end{align}
To verify, using the formula 3.4.28, which is\[ y\left ( x\right ) \thicksim cQ^{\frac{1-n}{2n}}\exp \left ( \omega \int ^{x}Q\left ( t\right ) ^{\frac{1}{n}}dt\right ) \] In this case, \(n=3\), since the ODE \(y^{\prime \prime \prime }=xy\) is third order. Here we have \(Q\left ( x\right ) =x\), therefore, plug-in into the above gives \begin{align} y\left ( x\right ) & \thicksim c\left ( x\right ) ^{\frac{1-3}{6}}\exp \left ( \omega \int ^{x}\left ( t\right ) ^{\frac{1}{3}}dt\right ) \nonumber \\ & \thicksim cx^{\frac{-1}{3}}\exp \left ( \omega \frac{4}{3}x^{\frac{4}{3}}\right ) \nonumber \end{align}
Comparing (1) and (2), we see they are the same.