3.9.6.0.5 Example 5 \(x^{2}y^{\prime \prime }+3xy^{\prime }+y=0\)

\begin{equation} x^{2}y^{\prime \prime }+3xy^{\prime }+y=0 \tag {1}\end{equation}

This is of course Euler ode, and we do not need to try this method as solving it as Euler ode is much simpler. But this is just for illustration for the case when the Liouville ode invariant comes out not a constant. In the form \(y^{\prime \prime }+p\left ( x\right ) y^{\prime }+q\left ( x\right ) y=r\left ( x\right ) \) then

\begin{equation} y^{\prime \prime }+\frac {3}{x}y^{\prime }+\frac {1}{x^{2}}y=0 \tag {1A}\end{equation}

Where now \(p=\frac {3}{x},q=\frac {1}{x^{2}},r=0\). Hence (6A) is

\begin{align*} z & =e^{-\int \frac {p}{2}dx}\\ & =e^{\frac {-3}{2}\int \frac {1}{x}dx}\\ & =\frac {1}{x^{\frac {3}{2}}}\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}{x^{2}}\right ) -\frac {1}{2}\left ( \frac {3}{x}\right ) ^{\prime }-\frac {1}{4}\left ( \frac {3}{x}\right ) ^{2}\\ & =\left ( \frac {1}{x^{2}}\right ) -\frac {3}{2}\left ( \frac {-1}{x^{2}}\right ) -\frac {1}{4}\left ( \frac {9}{x^{2}}\right ) \\ & =\frac {1}{x^{2}}+\frac {3}{2x^{2}}-\frac {9}{4x^{2}}\\ & =\frac {1}{4x^{2}}\end{align*}

Since \(q_{1}\) is not constant then the ode can not not converted to an ode in \(v\left ( x\right ) \) with constant coefficient.