# Definition:Variational Equation of Differential Equation

## Definition

Let

- $\map F {x,\langle\map {y^{\paren i} } x \rangle_{0\le i\le n} }=0,\quad n, i\in\N_0$

where $\langle \map {y^{\paren i} } x \rangle_{0\le i\le n}$ is a sequence of derivatives of 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 $F=0$.

## Sources

1963: I.M. Gelfand and S.V. Fomin: *Calculus of Variations* ... (previous) ... (next): $\S 5.27$: Jacobi's Necessary Condition. More on Conjugate Points