Definition:Variational Equation of Differential Equation

Definition
Let


 * $ F \left ( { x, \langle y^{ \left ( { i } \right ) } \left ( { x } \right )  \rangle_{ 0 \le i \le n}  } \right ) = 0, \quad n, i \in \N_{ 0 } $

where $ \langle y^{ \left ( { i } \right ) } \left ( { x } \right )  \rangle_{ 0 \le i \le n} $ is a sequence of derivatives of function $ y $, be a differential equation.

Let $ y \left ( { x } \right ), g \left ( { x } \right) $ be functions, which solve the given differential equation, such that


 * $ g \left ( { x } \right) = y \left ( { x } \right) + h \left ( { x } \right) $

Then, neglecting $ \mathcal O \left ( { h } \right )$, the differential equation satisfied by $ h $ is called the variational equation of the differential equation $ F = 0 $.