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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 $\mu^i$?

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    $\begingroup$ This is not an ODE, but rather an integro-differential equation. Though, with the substitution $e_i(t) = \partial f_i(t)/\partial t$ it could be reduced to a second order ODE in $f_i(t)$ (provided $f_i(0)=0$). It is also best to assume that there might not be any explicit and general solution. Rather, better to think about what properties would you like to extract from he solution if you had it? Such questions might be answerable even without finding an explicit solution first. $\endgroup$
    – Igor Khavkine
    Commented Jul 31, 2016 at 6:54
  • $\begingroup$ Unfortunately, I do need an explicit solution, this came up in a computational problem and yes I implicitly thought of the substitution you mentioned; thanks. $\endgroup$
    – ABIM
    Commented Jul 31, 2016 at 15:22
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    $\begingroup$ Is there any reason your "explicit" solution can not be numerical? $\endgroup$
    – Mark L. Stone
    Commented Jul 31, 2016 at 19:33
  • $\begingroup$ Yes because I need to solve for $\sigma$ after obtaining the solution for $e_i$. $\endgroup$
    – ABIM
    Commented Jul 31, 2016 at 21:34
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    $\begingroup$ You haven't specified what it means for you to "solve for $\sigma$", but if you are attacking this problem numerically, there's probably a way to do that part numerically as well. $\endgroup$
    – Igor Khavkine
    Commented Jul 31, 2016 at 22:58

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