3.9.5 Transformation on the dependent variable (method 1) \(y=v\left ( x\right ) z\left ( x\right ) \)

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.