In my research i came across this certain ODE and I've reduced it to this form: \begin{equation} \sum_{i=1}^N c_i \frac{\partial e_i(t)}{\partial t} + \sum_{i,j=1}^N \sigma^i(\sigma^j)^te_i(t)\int_0^t e_j(s) ds - \sum_{i=1}^N e_i(t)\mu_i=0, \end{equation} where $\mu_i(t),\sigma^i(t)$ are pre-prescribed $C^2$ functions.

How can I solve this ODE for $e_i(t)$ in therms of the $\sigma^i$ and $\mu^i$?