Let \(f\) be an isometric immersion of a pseudo-Riemannian manifold \(M^n_q\) in a nonflat pseudo-Riemannian space form \(M^{n+1}_p\) such that \(\Delta f= Af + B\), where \(A\) is an endomorphism of \(\mathbb{R}^{n+2}_r\) and \(B\) is a constant vector. The authors prove that the above algebraic condition is fulfilled if and only if \(M^n_q\) is an open piece of one of the following hypersurfaces of \(M^{n+1}_p\): (1) a minimal hypersurface, (2) a totally umbilical hypersurface, (3) a product of two nonflat totally umbilical submanifolds or (4) a special class of quadratic hypersurfaces.