calculus of variations - Basis for solution space of Jacobi accessory equation

The Jacobi accessory equation has importance as a means of checking candidates for functional extrema. A book of mine ($\textit{Calculus of variations}$, 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 $c_1, c_2$, then the functions
$$u_1(x) = \frac{\partial y}{\partial c_1}, \quad u_2(x) = \frac{\partial y}{\partial c_2}$$

evaluated at some particular $(c_1, c_2)$ are solutions to the Jacobi accessory equation (given basic smoothness assumptions). However, van Brunt goes on to claim without proof that $u_1, u_2$ $\textit{form a basis for the solution space}$. Can anyone suggest how this might be proved?