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How do we solve a a system of ODE's:

$\frac{dc_i}{dx} = f_i (c_i, x) $

Where $f_i $ can be linear or non-linear. Subject to: $\sum{\alpha_i c_i} = 0$?

More specifically, this is the system of equations I'm looking at:

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    In general this system seems to be over determined so that one may not expect any solution. Can you provide more details about the problem? – 2017-01-11
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    I posted details of the system I'm looking at. – 2017-01-11

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The system had better be consistent with the constraint, i.e. (assuming $\alpha_i$ are constants) $$\dfrac{d}{dx} \sum_i \alpha_i c_i = \sum_i \alpha_i f_i(c_i, x) = 0$$ If so, just solve the constraint for one of the $c_i$ and substitute in to the differential equations for the other $c_i$'s.

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    This was my thinking. I have posted more info about the equation. I'm confused that they already used Eqn (21) to derive Eqn (22), but they state that Eqn (21) is used with to solve the system? – 2017-01-11
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    "... similar to that for the DSPM". What did they do for the DSPM? – 2017-01-11