Consider the space of twice continuously differentiable functions \(C^{2}=\{f(x), x \in[0,1]\}\). Solve the functional differential equation

\[\frac{\delta F[f]}{\delta f(x)}=\left[-\frac{d^{2} f}{d x^{2}}+c f(x)+g f^{3}(x)\right] F[f]\]

where \(c, g\) are constants, for the functional \(F[f]: C \rightarrow R^{1}\).