3.9.6.0.7 Example 7 \(y^{\prime \prime }-\frac {1}{\sqrt {x}}y^{\prime }+\left ( \frac {1}{4x}+\frac {1}{4x^{\frac {3}{2}}}-\frac {2}{x^{2}}\right ) y=0\)

\begin{equation} y^{\prime \prime }-\frac {1}{\sqrt {x}}y^{\prime }+\left ( \frac {1}{4x}+\frac {1}{4x^{\frac {3}{2}}}-\frac {2}{x^{2}}\right ) y=0 \tag {1}\end{equation}

In the form \(y^{\prime \prime }+p\left ( x\right ) y^{\prime }+q\left ( x\right ) y=r\left ( x\right ) \) then \(p=-\frac {1}{\sqrt {x}},q=\left ( \frac {1}{4x}+\frac {1}{4x^{\frac {3}{2}}}-\frac {2}{x^{2}}\right ) ,r=0\). Hence (6A) is

\begin{align*} z & =e^{-\int \frac {p}{2}dx}\\ & =e^{\int \frac {1}{\sqrt {x}}dx}\\ & =e^{2\sqrt {x}}\end{align*}

Now we check if Liouville ode invariant \(q_{1}\) is constant.

\begin{align*} q_{1} & =q-\frac {1}{2}p^{\prime }-\frac {1}{4}p^{2}\\ & =\left ( \frac {1}{4x}+\frac {1}{4x^{\frac {3}{2}}}-\frac {2}{x^{2}}\right ) -\frac {1}{2}\left ( -\frac {1}{\sqrt {x}}\right ) ^{\prime }-\frac {1}{4}\left ( -\frac {1}{\sqrt {x}}\right ) ^{2}\\ & =-\frac {2}{x^{2}}\end{align*}

Not constant. Stop here. This can be solved using Kovacic algorithm.