Transformation on the independent variable \(x\) method 2 ode internal name "second_order_change_of_variable_on_x_method_2"

Given ode\begin {equation} y^{\prime \prime }+p\left ( x\right ) y^{\prime }+q\left ( x\right ) y=r\left ( x\right ) \tag {A} \end {equation} Let \(\tau =g\left ( x\right ) \) where \(\tau \) is the new independent variable. Applying this to (A) results in (details not shown)\begin {equation} y^{\prime \prime }\left ( \tau \right ) +p_{1}\left ( \tau \right ) y^{\prime }\left ( \tau \right ) +q_{1}\left ( \tau \right ) y\left ( \tau \right ) =r_{1}\left ( \tau \right ) \tag {1} \end {equation} Where \begin {align} p_{1}\left ( \tau \right ) & =\frac {\tau ^{\prime \prime }\left ( x\right ) +p\left ( x\right ) \tau ^{\prime }\left ( x\right ) }{\left ( \tau ^{\prime }\left ( x\right ) \right ) ^{2}}\tag {2}\\ q_{1}\left ( \tau \right ) & =\frac {q\left ( x\right ) }{\left ( \tau ^{\prime }\left ( x\right ) \right ) ^{2}}\tag {3}\\ r_{1}\left ( \tau \right ) & =\frac {r\left ( x\right ) }{\left ( \tau ^{\prime }\left ( x\right ) \right ) ^{2}} \tag {4} \end {align}

The idea of the transformation is to determine if ode (1) can be solved instead of (A).

Let \(p_{1}=0\) then \(\tau \) is solved for from \(\tau ^{\prime \prime }\left ( x\right ) +p\left ( x\right ) \tau ^{\prime }\left ( x\right ) =0\). \[ \tau =\int e^{-\int pdx}dx \] If this solution \(\tau \left ( x\right ) \) results in \(q_{1}\) above being a constant, then (1) can now be easily solved.