In this paper, we classify timelike hypersurfaces in Lorentz-Minkowski space, x : M_1^n-----> L^{n+1}, satisfying the condition L_k x = Ax + b, where L_k is the kth extension of Laplace operator (i.e. ), A is a constant matrix and b is a constant vector. The condition L_kx = Ax + b is a new version of a well-known equation x = dx for a real number d. As an extension of Takahashi's theorem we show that such a hypersurface has to be k-minimal or an open piece of S_1^n(c), S_1^m \times (c) R^{n-m} or S^m(c) \times L^{n-m} for some c > 0 and 1 < m < n.
