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the branch of mathematics that deals with the mathematical aspects of problems from science and engineering: applied analysis, numerical mathematics, applied statistics etc. (For applications of mathematics in general, cf. also the [applications] tag.)
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Fit a system of linear ODEs from several experiments
(This is more of a comment but not enough reputation to do)
It seems a case for system identification (e.g. http://es.mathworks.com/products/sysid/) but I do not get the dependence on $p$, does it st …