+ + =

This equation is for a system of three moving masses (three second-order equations lead to one sixth-order equation). All the masses, springs, and dampers in this system had to be given numerical values. It is impossible to solve such a problem with letter values. Once we assign numerical values, the root finder makes our job quick and "painless." For the MATLAB root finder we would just write

In any problem, once we have the roots (either in letter form or number form), we immediately write down the complementary solution, using a set of rules proven

in courses on differential equations, but which we here just accept "on faith." If, in a math course, you were taken through some intermediate steps at this point, I urge you to now ignore these needlessly time-consuming operations. That is, as engineers, when a mathematical process is nothing but a routine (dare I say cookbook!), we treat it as such and spend our time on more creative aspects of our work.