Change of variables and Rewriting an ODE

Question

Given the second order ODE: \begin{equation} y'' + yy' + xy^4 = 0 \end{equation}

with a change of variables of: \begin{align} r & = xy & s& = \ln x \\ y & = re^{-s} & x & = e^s\\ \end{align}

the example I am looking at then states the ODE can be rewritten as: \begin{equation} s_{rr} = (r^4 - r^2 +2r)s^3_r + (r-3)s^2_r \end{equation}

I am battling to fill in the blanks as to how they arrive at that answer.

Answer 1

Below a way for the change of variables :