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There is really no complication in arriving at equation (5) given equation (5). We have: $$\frac{d}{d\rho}\left[\frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}\right]=0.$$ We solve this differential equation. $$\frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}=\tilde{c}$$ $\tilde{c}$ being a ...