While reviewing an article "A note on the wall-jet problem" by MERKIN and NEEDHAM, I found one integration a bit annoying. Need a little help to sort it out.
We have an equation \begin{equation} \int_{0}^{\infty} \left(f(x)-\alpha\right)f'^2(x) dx=1 \end{equation} where \begin{align} f^{'}(x)=\frac{1}{6}f^{1/2}(x)\left(\sigma^3-f^{3/2}(x)\right) \end{align} for $\sigma^2=f(\infty).$\ The BCs for the problem are $f(0)=\alpha,$ $f^{'}(0)=\beta$ and $f^{'}(\infty)=0.$
The question is how to get the following equation from the above intergral? \begin{equation} \sigma^{8}-\frac{20}{9}\alpha\sigma^{6}+\frac{16}{9}\alpha^{5/2}\sigma^{3}-\left(40+\frac{5\alpha^4}{9}\right)=0 \end{equation}
