Erreur

Comme ça ?
\begin{equation}
\begin{aligned}
\partial_tu(t,\xi) = &\partial_{\xi}\big(c(\xi)\partial_{\xi}u(t,\xi)\big)-p(\xi)u(t,\xi)+f\big(u(t,\xi))\big)-w(t,\xi)\\ &{}+\partial_t\beta_1(t,\xi)\;, \qquad t \ge 0\;, \qquad \xi \in [0,1] \\

\partial_tw(t,\xi) = &-\alpha w(t,\xi)+\gamma u(t,\xi)+\partial_t\beta_2(t,\xi)\;, \\ & \phantom{{}+\partial_t\beta_1(t,\xi)\;,{}}\qquad t \ge 0\;, \qquad \xi \in [0,1]
\end{aligned}

\end{equation}