When differential equations are being developed, what dictates the order of the differential? What decides if it is second order, third order, fourth order, fifth order, etc... I understand that the process is taking the derivative of a derivative of a derivative, but what decides if the the third, fourth, or fifth derivative has any value when performing the analysis of process being investigated?
It depends upon what the derivative is supposed to represent. "Dynamics" problems, about motion, depend upon "F= ma" and acceleration, a, is the second derivative of the position function so if the problem is to determine position from a given force function, the differential equations are typically second order. On the other hand, problems involving just "growth" or "rate of flow", because those are just "rate of change", tend to be first order differential equations. Problems involving "elasticity", on the other hand, but again from physical reasons tend to be fourth order equations.