The Jacobi(f, t, x(t), X(t), h, a) command finds Jacobi's equation and tries to find solutions of Jacobi's equation, that is, conjugate points.

The routine returns an expression sequence consisting of Jacobi's equation and any solutions found by dsolve.

If dsolve encounters a problem, an error message is returned.

If dsolve fails to find a solution, only Jacobi's equation is returned.

If the solution of Jacobi's equation has a zero on the region of interest, the extremal is not optimal.

$\mathrm{with}\left(\mathrm{VariationalCalculus}\right)$

$\left[\mathrm{ConjugateEquation}\,\mathrm{Convex}\,\mathrm{EulerLagrange}\,\mathrm{Jacobi}\,\mathrm{Weierstrass}\right]$

$f≔-\frac{{\mathrm{diff}\left(y\left(t\right)\,t\right)}^2}{2}+\frac{{y\left(t\right)}^2}{2}$

$f≔-\frac{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}{2}+\frac{{y\left(t\right)}^2}{2}$

$\mathrm{Jacobi}\left(f\,t\,y\left(t\right)\,\sin \left(t\right)\,h\,0\right)$

$\frac{{\ⅆ}^2}{\ⅆt^2}h\left(t\right)+h\left(t\right),h\left(t\right)=\mathrm{c\_\_1}\sin \left(t\right)$