conjugate_equation takes four arguments :
The function y_{0}(x) is assumed to be stationary for the problem of minimizing some functional F(y)=∫_{a}^{b}f(x,y,y′) dx. The return value is the expression

 − 

 , (4) 
at α=A and β=B, which is zero if and only if t is conjugate to a. To find any conjugate points, set the returned expression to zero and solve.
For example, we find a minimum for the functional
F(y)=  ∫ 
 ⎛ ⎝  y′(x)^{2}−x y(x)−y(x)^{2}  ⎞ ⎠  dx 
on D={y∈ C^{1}[0,π/2]:y(0)=y(π/2)=0}. The corresponding EulerLagrange equation is :
^
2x*y(x)y(x)^
2,y(x))
Output :
The general solution is :
Output :
The stationary function depends on two parameters c_{0} and c_{1} which are fixed by the boundary conditions :
Output :
Input :
Output :
The above expression obviously has no zeros in (0,π/2], hence there are no points conjugate to 0. Since f_{y′ y′}=2>0, where f(y,y′,x) is the integrand in F(y) (the strong Legendre condition), y_{0} minimizes F on D. To obtain y_{0} explicitly, input :
Output :