Jacobi Differential Equation (Calculus of Variations)

$${d\over dx} \Omega_{\eta'}-\Omega_\eta={d\over dx} (f_{y'y}\eta+f_{y'y}\eta')-(f_{yy}\eta+f_{yy'}\eta')=0,$$

where

$$\Omega(x,\eta,\eta')\equiv {\textstyle{1\over 2}}(f_{yy}\eta^2+2f_{yy'}\eta\eta'+f_{y'y}\eta'^2).$$

This equations arises in the Calculus of Variations.

References

Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162-163, 1925.