Multline inside Align

Environments which end with an ed will not change the mode to display math. So you can use them inside a math environment like align, equation, gather. For example, you can use multlined inside align.

\begin{align*}
	\sigma_{k+1}(t) &=
		\begin{multlined}[t]
		  - \frac{c_k}{c_{k+1}} \frac{1}{t} (\xi_{k+1} + \eta_{k+1} \tau(1-\omega t)) q_k(t) + \frac{1}{t} q_k(t)\\+\left(\frac{c_k}{c_{k+1}}(\xi_{k+1} + \eta_{k+1} \tau) - 1\right) \frac{1}{t} q_{k-1}(t)
		\end{multlined}\\
	&= \frac{c_k}{c_{k+1}} \eta_{k+1} \omega \tau q_k(t) + \left(\frac{c_k}{c_{k+1}}(\xi_{k+1} + \eta_{k+1} \tau) - 1\right) \sigma_k(t)
\end{align*}

$$ \require{mathtools} \begin{aligned} \sigma_{k+1}(t) &= \begin{multlined}[t] - \frac{c_k}{c_{k+1}} \frac{1}{t} (\xi_{k+1} + \eta_{k+1} \tau(1-\omega t)) q_k(t) + \frac{1}{t} q_k(t)\\+\left(\frac{c_k}{c_{k+1}}(\xi_{k+1} + \eta_{k+1} \tau) - 1\right) \frac{1}{t} q_{k-1}(t) \end{multlined}\\ &= \frac{c_k}{c_{k+1}} \eta_{k+1} \omega \tau q_k(t) + \left(\frac{c_k}{c_{k+1}}(\xi_{k+1} + \eta_{k+1} \tau) - 1\right) \sigma_k(t) \end{aligned} $$

To use multlined you will need \usepackage{mathtools}. The other ed versions are included in amsmath I think.