jacobi_equation takes five or six arguments :

- expression f(y,y′,x),
- independent variable x,
- dependent variable y (this argument and the previous one can be combined to a single argument y(x), in which case the call has five arguments),
- expression y
_{0}∈ C^{1}[a,b] which is stationary for the functional F(y)=∫_{a}^{b}f(y,y′,x) dx, - symbol h for the unknown function in Jacobi equation,
- point a∈ℝ, which is the lower bound for x.

The return value contains the Jacobi equation

− |
| ⎛ ⎝ | f_{y′ y′}(y_{0},y_{0}′,t) h′ | ⎞ ⎠ | + | ⎛ ⎜ ⎜ ⎝ | f_{y y}(y_{0},y_{0}′,t)− |
| f_{y y′}(y_{0},y_{0}′,t) | ⎞ ⎟ ⎟ ⎠ | h=0. (3) |

If the Jacobi equation has a solution such that h(a)=0, h(c)=0 for some c∈(a,b] and h not identically zero on [a,c], then y_{0} does not minimize the functional F. It is said that c is *conjugate* to a. The function y_{0} minimizes F if f_{y′ y′}(y_{0},y_{0}′,t)>0 for all t∈[a,b] and there are no points conjugate to a in (a,b].

If the Jacobi equation can be solved by dsolve, a sequence containing the equation (3) and its solution is returned. Otherwise, if (3) cannot be solved immediately, only the Jacobi equation is returned.

For example, input :

jacobi_equation(-1/2*y’(t)

`^`

2+y(t)`^`

2/2,t,y,sin(t),h,0)
Output :

(-diff(h(t),t,2)-h(t))=0, c_0*sin(t)