### 4.3 note p1 added 2/3/2017

4.3.1 problem (a), page 88
4.3.2 problem (b), page 88
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Current rules I am using in simpliﬁcations are

1.
$$S_{0}^{\prime }\ggg S_{1}^{\prime }$$
2.
$$S_{0}^{\prime }S_{1}^{\prime }\ggg \left ( S_{1}^{\prime }\right ) ^{2}$$
3.
$$S_{0}\ggg S_{1}$$
4.
$$\left ( S_{0}\right ) ^{n}\ggg S_{0}^{\left ( n\right ) }$$ (the ﬁrst is power, the second is derivative order).

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 $$\left ( S^{\prime }\right ) ^{2}$$?

#### 4.3.1 problem (a), page 88

$y^{\prime \prime }=\frac{1}{x^{5}}y$ 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.

#### 4.3.2 problem (b), page 88

$y^{\prime \prime \prime }=xy$ 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*}

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}