The Difficulty of Solving Equations

The following comes from General System Theory by Ludwid von Betalanffy, 1968:

Sets of simultaneous differential equations as a way to "model" or define a system are, if linear, tiresome to solve even in the case of a few variables; if nonlinear, they are unsolvable except in special cases (Table 1.1).

Table 1.1
Classification of Mathematical Problems *
and Their Ease of Solution By Analytical Methods

After Franks, 1967

. . . Linear . . . . Nonlinear .
Equation Type: One
Equation		 . Several
Equations		 . Many
Equations		 . One
Equation		 Several
Equations		 Many
Equations
Algebraic Trivial . Easy . Essentially Impossible . Very Difficult Very Difficult Impossible
Ordinary Differential Easy . Difficult . Essentially Impossible . Essentially Impossible Impossible Impossible
. . . x . . . . . .
Partial Differential Difficult . Essentially Impossible . Essentially Impossible . Impossible Impossible Impossible

* Courtesy of Electronic Associates, Inc.

For this reason, computers have opened a new approach in systems research; not only by way of facilitation of calculations which otherwise would exceed available time and energy and by replacement of mathematical ingenuity by routine procedures, but also by opening up fields where no mathematical theory or ways of solution exist. Thus systems far exceeding conventional mathematics can be computerized; on the other hand, actual laboratory experiment can be replaced by computer simulation, the model so developed then to be checked by experimental data. In such a way, for example, G. Hess has calculated the fourteen-step reaction chain of glycolysis in the cell in a model of more than 100 differential equations. Similar analyses are routine in economics, market research, etc.