In the present work it is studied the initial value problem for an equation of the form L ∂ k u ∂ t k = ∑ j = 1 k L j ∂ k − j u ∂ t k − j , where L is an elliptic partial differential operator and ( L j : j = 1 , … , k ) is a family of partial differential operators with bounded operator coefficients in a suitable function space. It is found a suitable formula for solution. The correct formulation of the Cauchy problem for this equation is also studied.