Reference post: click here
Given, \begin{eqnarray} &&\Delta p_5-p_5+3S^2p_5 +\frac{SZ}{576\sqrt{\lambda}}(3Z-5S^3) \left(\frac{15g_5}{\lambda^2}+1\right)^2\nonumber\\ &&+\frac{S^3}{32\sqrt{\lambda}}\left[(\nabla S)^2-S^2\right] -\frac{S^7}{576\sqrt{\lambda}}\left(\frac{315g_7}{\lambda^3} -\frac{60g_5}{\lambda^2}+1\right)=0 \,. \end{eqnarray} Where, $\Delta$ is double derivative.
I need to evaluate the solution for $p_5$, so the equation \begin{eqnarray} &&\phi_5=p_5 +\frac{S^5}{1152\sqrt{\lambda}}\left(\frac{3g_5}{\lambda^2}+2\right) \nonumber\\ &&-\frac{S}{384\sqrt{\lambda}}\left[ \left(\frac{30g_5}{\lambda^2}+2\right)SZ+12S^2-12(\nabla S)^2 -\left(\frac{15g_5}{\lambda^2}-2\right)S^4 \right] \,. \end{eqnarray} will take this form,
$$\phi_5=\frac{1}{9}\sqrt{\frac{2}{3}}\Biggl[ Y-\frac{2275}{64}S^2Z+\frac{1503}{16}Z -\frac{15}{32}S\left(\frac{dS}{d\rho}\right)^2 -24S-\frac{595}{96}S^3+\frac{11285}{384}S^5 \biggr]$$ Where $g_3= -\frac{3}{2}, g_5= 0, \lambda= \frac{3}{2}$
In the paper, they don't actually solve the equation for $p_5$, or for $p_1$ or $p_3$, for that matter. In each case they introduce a new variable that allows them to manipulate equations. So, for instance, $S$ is defined by the equation $S=p_1 \sqrt{\lambda}$. We have not actually solved $p_1$, we're just defining $S$. Similarly, $Z$ is defined by the equation $p_3=\frac{\sqrt{2}}{3\sqrt{3}}(\frac{65}{8}Z-\frac{8}{3}S-\frac{19}{12}S^3)$.
Similarly, $p_5$ is not actually solved; instead, equation 47 is used to define a new variable $Y$ that has never appeared before. So the form they give for $\phi_5$ is obtained simply by plugging in equation 47.