The Jacobi accessory equation has importance as a means of checking candidates for functional extrema. A book of mine (Calculus of variations, by van Brunt) proves that we can find solutions to the Jacobi accessory equation by differentiating the general solution to the Euler-Lagrange equation; that is, if the latter has a general solution y involving parameters c1,c2, then the functions
u1(x)=∂y∂c1,u2(x)=∂y∂c2
evaluated at some particular (c1,c2) are solutions to the Jacobi accessory equation (given basic smoothness assumptions). However, van Brunt goes on to claim without proof that u1,u2 form a basis for the solution space. Can anyone suggest how this might be proved?
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