Definition:Variational Equation of Differential Equation

Definition
For $n, i \in \N$, let:


 * $\map F {x, \sequence {\map {y^{\paren i} } x}_{0 \mathop \le i \mathop \le n} } = 0$

where $\sequence {\map {y^{\paren i} } x}_{0 \mathop \le i \mathop \le n}$ is a sequence of derivatives of a function $y$, be a differential equation.

Let $\map y x, \map g x$ be real functions which solve the given differential equation, such that


 * $\map g x = \map y x + \map h x$

Then, neglecting $\map {\mathcal O} {h^2}$, the differential equation satisfied by $h$ is called the variational equation of the differential equation $F = 0$.