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In Mukhanov's "Physical foundations of cosmology" on page 102 the author considers an equation for the evolution of the neutron mass fraction $X_{n}\equiv n_{n}/(n_{p}+n_{n})$: $$ \tag 1 \dot{X}_{n}(t) = -\lambda_{n\to p}(t)(1+e^{-Q/T(t)})(X_{n} - X_{n,\text{eq}}), \quad X_{n}(t\to 0) = X_{n,\text{eq}}(t), $$ where

  • $\lambda_{n\to p}$ the neutron-to-proton conversion rate (behaving as $\lambda_{n\to p} \sim t^{-\alpha}$, where $\alpha > 0$),
  • $Q = m_{n}-m_{p}$,
  • $T(t)$ is the temperature (behaving as $T \sim 1/\sqrt{t}$),
  • $X_{n,\text{eq}}(t)$ is the neutron mass fraction in equilibrium, $$ X_{n,\text{eq}}(t)= \frac{1}{1+e^{Q/T(t)}}, \quad X_{n,\text{eq}}(0) = \frac{1}{2} $$ The formal solution of $(1)$ is $$ \tag 2 X_{n}(t) = X_{n,\text{eq}}(t) - \int \limits_{0}^{t}d\tilde{t} \dot{X}_{n,\text{eq}}(\tilde{t})\exp\left[-\int \limits_{\tilde{t}}^{t}d\bar{t}\lambda_{n\to p}(\bar{t})(1+e^{-Q/T(\bar{t})})\right] $$ The author says that for small enough $t$ the second term tends to zero and can be considered as a perturbation. Namely, he says that it is possible to write $(2)$ as the asymptotic series in increasing powers of $\dot{X}_{n,\text{eq}}$ using the integration by parts: $$ \tag 3 X_{n}(t) = X_{n,\text{eq}}(t)\left[ 1-\frac{1}{\lambda_{n\to p}(t)(1+e^{-Q/T(t)})}\frac{\dot{X}_{n,\text{eq}}(t)}{X_{n,\text{eq}}(t)}+\dots\right] $$ I absolutely do not understand how to obtain $(3)$...

Could you please help me?

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