Definition:Formation of Ordinary Differential Equation by Elimination

Definition
Let $\map f {x, y, C_1, C_2, \ldots, C_n} = 0$ be an equation:
 * whose dependent variable is $y$
 * whose independent variable is $x$
 * $C_1, C_2, \ldots, C_n$ are constants which are deemed to be arbitrary.

A differential equation may be formed from $f$ by:
 * differentiating $n$ times $x$ to obtain $n$ equations in $x$ and $\dfrac {\d^k y} {\d x^k}$, for $k \in \set {1, 2, \ldots, n}$
 * eliminating $C_k$ from these $n$ equations, for $k \in \set {1, 2, \ldots, n}$.