The given functional was $\ J[f] =\int_{0}^{1} ({x^{2}-f^{2}(x) + (f'(x))^{2}})dx $. Using the Euler-Lagrange equation, I calculated that $\ L_f=-2f$, $\ L_{f'}=2f'$, and $\ \frac{d}{dx}(L_{f'})=2f'' $.
Such that $\ 0 = L_f - \frac{d}{dx}(L_{f'})= -2f-2f''$ for which the solution is: $\ f(x)=c_2sin(x) + c_1cos(x)$
Functionals are still new to me, so any help is appreciated.