In the first, the initial concentrations of A and B are the same. The rate of buildup of C looks like the expression for the buildup of B in equation 5 since both A and B are consumed at the same rate. If the reverse is the case, then interchange all the A's and B's, leaving the C's alone. By now I'm sure you have the impression that if you like differential equations, you'll love kinetics. This is indeed the case.
Numerical methods have to be used.
The answer is that one can set up the situation so that for many practical purposes, the complications go away. Take the second order reaction, for example. Pseudo-first order reactions are quite commonly used for quantification. We use our selection of reactants to get selectivity, then we load up the system with one reactant so we get a reaction rate proportional to the concentration of the other reactant. Presuming we can keep conditions constant with respect to everything except the concentration of one, low-concentration species, the reaction rate is proportional to the concentration of that species and we have a means of getting a linear working curve in that species.