Formation of Differential Equations

Differential equations that model real problems do not, in practice, arise by starting from a family of curves and eliminating constants — they arise directly from a physical law or geometric condition. Even so, the reverse process — forming a differential equation from a family of curves by eliminating its arbitrary constants — is an essential skill, both because it shows precisely which differential equation a given family of solutions satisfies, and because the technique (successive differentiation, then elimination) is the standard way of checking or discovering the order of the equation a family belongs to.

The general elimination method. Given an equation for a family of curves containing $n$ arbitrary constants:

1. Differentiate the equation successively $n$ times, producing $n$ new equations (so $(n+1)$ equations in total, counting the original).
2. Eliminate the $n$ arbitrary constants using these $(n+1)$ equations.
3. The result — necessarily containing a derivative of the $n$th order — is the required differential equation. …