ode internal name "second_order_change_of_variable_on_y_method_1"
This is also called Liouville transformation. Book by Einar Hille, ordinary differential
equations in the complex domain. Page 179. This method assumes that
\[ y=v\left ( x\right ) z\left ( x\right ) \]
Substituting this
into (A) results in the following ode where the dependent variable is
\(v\) and not
\(y\)\begin{equation} v^{\prime \prime }\left ( x\right ) +\left ( p+\frac {2}{z}z^{\prime }\left ( x\right ) \right ) v^{\prime }\left ( x\right ) +\frac {1}{z}\left ( z^{\prime \prime }\left ( x\right ) +pz^{\prime }\left ( x\right ) +qz\left ( x\right ) \right ) v\left ( x\right ) =\frac {r}{z} \tag {6}\end{equation}
Assuming
that coefficient of
\(v^{\prime }\) in (6) zero implies
\[ p+\frac {2}{z}z^{\prime }\left ( x\right ) =0 \]
Solving gives (where constant of integration is taken
as one)
\begin{equation} z=e^{-\int \frac {p}{2}dx} \tag {6A}\end{equation}
With this choice (6) becomes
\[ v^{\prime \prime }+\frac {1}{z}\left ( z^{\prime \prime }+pz^{\prime }+qz\right ) v=\frac {r}{z}\]
Substituting
\(z\) from (6A) into the above reduces it to
(after some algebra) to
\begin{equation} v^{\prime \prime }+q_{1}v=r_{1} \tag {6B}\end{equation}
Where
\begin{align*} q_{1} & =q-\frac {1}{2}p^{\prime }-\frac {1}{4}p^{2}\\ r_{1} & =\frac {r}{z}\\ & =re^{\frac {1}{2}\int pdx}\end{align*}
\(q_{1}\) is called the Liouville ode invariant. If \(q_{1}\) is constant, or constant divided by \(x^{2}\),
then the substitution \(y=\) \(v\left ( x\right ) z\left ( x\right ) \) used in the original original ode results in a constant
coefficient ode. In \(y=\) \(v\left ( x\right ) z\left ( x\right ) \) the \(z\left ( x\right ) \) term is known from 6A and \(v\left ( x\right ) \) is the new unknown dependent
variable.
The new ode will be in \(v\left ( x\right ) \) but with constant coefficients. Solving it for \(v\left ( x\right ) \) gives \(y\). Examples
given below to illustrate this method.