# Definition:Variational Equation of Differential Equation

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## 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 $F = 0$.

## 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