I am trying without success to understand how two formulae in appendix B of this paper are derived.
Equation B1 is an equation for perturbations, obtained from regular perturbation theory: $$\ddot{\delta} + 2\frac{\dot{a}}{a}\dot{\delta} = 4\pi G \rho \delta$$ where the dot means derivative with respect to time $t$.
I'm happy with equations B2 and B3. which are Friedman equations in general relativity for a cosmology where the universe is expanding, and I don't have any problems with B4 to B6. But I don't understand how B1 (above), B5 and B6: $$\begin{eqnarray} \frac{\ddot{a}a}{\dot{a}^2} = -\frac{1}{2}-\frac{3}{2}w(1 - \Omega)\\ \frac{d\Omega}{d \ln a} = 3w(1 - \Omega)\Omega \end{eqnarray}$$ allow us to derive B7: $$\frac{d^2 \ln \delta}{d \ln a^2} + \left(\frac{d \ln \delta}{d \ln a}\right)^2 + \frac{d \ln \delta}{d \ln a}\left[\frac{1}{2} - \frac{3}{2}w(1 - \Omega)\right] = \frac{3}{2}\Omega$$
or how defining $\alpha$ as $\Omega^\alpha \equiv \frac{d \ln \delta}{d \ln a}$ allows us to combine B6 and B7 to get B10: $$3w(1-\Omega)\Omega\ln\Omega\frac{d\alpha}{d\Omega} - 3w\left(\alpha-\frac{1}{2}\right)\Omega + \Omega^\alpha - \frac{3}{2}\Omega^{1-\alpha} + 3w\alpha - \frac{3}{2}w + \frac{1}{2} = 0$$
What am I missing?